User tom copeland - MathOverflow

The Dedekind Eta Function in Physics

2012-12-03T01:26:57Z

This interesting little fellow popped up in some operator algebra (Witt / Virasoro Lie algebra) I was exploring, so I've been curious about where else the Dedekind $\eta$-function makes a cameo appearance and found that in physics it's related to the

1) statistical parameters of solvable Ising models

(See "The Reasonable and Unreasonable Effectiveness of Number Theory in Statistical Mechanics" by G. Andrews and "Introduction to Exactly Solvable models in Statistical Mechanics" by C. Tracy.)

The difference between the average local occupation densities of two sub-lattices of a hard hexagon model of a lattice gas given on pp. 368-371 of Tracy is $R(\tau)=\frac{n(\tau)\eta(5\tau)}{\eta^2(3\tau)}$.

2) partition functions (statistical mechanics variety) for colored bosons moving on a line (1/24 is the associated Casimir energy) and one-color fermions

3) operator traces (characters) for the infinite dimensional Lie algebras $\widehat{su}_n$, equivalent to 2-Dim current algebras

4) partition function of a microscopic black-hole in a 5-Dim D-brane

5) string theory guage corrections

(For 2-5, see "Nucleon Structure, Duality and Elliptic Theta Functions" by W. Scott. For item 2, see also "Vertex Operators and Modular Forms" by G. Mason and M. Tuite.)

From pg. 39 of "Fivebrane instantons ..." and on pg. 11 of "D3 instantons ...," a correction to the field basis (of the RR axion dual to D3-branes) in type IIB string perturbation theory related to the action of S-duality in ten dimensions:

$\tilde{c_a} \mapsto \tilde{c_a}-\tilde{c}_{2,a}\:\epsilon(g)$ where, with $g=\binom{a\:\:b}{c\:\:d}$,

$$\exp(2\pi i \epsilon(g))=\frac{\eta\left [ \frac{a\tau+b}{c\tau+d} \right ]}{\left ( c\tau+d \right )^{\frac{1}{2}}\eta(\tau)}.$$

6) partition function in 2+1 dimensions and vanishing chemical potential of non-relativistic fermions in a constant magnetic field

("Nonrelativistic Fermions in Magnetic Fields: a Quantum Field Theory Approach" by O. Espinosa, J. Gamboa, S. Lepe, and F. Mendez)

7) physics of gauge theories and the Dirac operator

(See "The Logarithm of the Dedekind $\eta$ Function" by M. Atiyah.)

Michael Atiyah even goes so far as to say, "It seems therefore timely to attempt to survey the whole development of the theory of $\log(\eta)$, putting results in their natural order and in the appropriate general context. This is the aim of the present paper, in which the emphasis will be strongly geometrical. In a sense we shall show that the latest ideas from physics [circa 1987] provide the key to a proper understanding of Dedekind's original results."

8) knots and dynamics

(See "Knots and Dynamics" by E. Ghys, and Chapter 2 A New Twist in Knot Theory in Dana MacKenzie's book What's Happening in the Mathematical Sciences Vol. 7.)

Ghys presents the equation $$24\log\eta\left(\frac{a\tau+b}{c\tau+d}\right)=24\: log(\eta(\tau))+6\: log(-(c\tau+d)^{2})+2\pi i\:\mathfrak{R}\left(\binom{a\: b}{c\: d}\right) $$

where $\mathfrak{R}$ is the Rademacher function, which he relates to the linking number

between two knots related to modular/Lorenz flow: “For every hyperbolic element $A=\binom{a\: b}{c\: d}$ in $PSL(2,Z)$, the linking number between the [modular/Lorenz] knot $k_A$ and the trefoil knot $l$ is equal to $\mathfrak{R}(A)$ ....”

9) string/brane partition functions, propagators, and metrics

In "String Theory" by S. Nibbelink, $\eta$ occurs in the denominator of string partition functions for fermionic and bosonic zero modes (pp. 163-7).

A coefficient in the 10-dim metric for a 7-brane is given as $e^{\phi}=\tau_2 \eta^2\bar{\eta}^2|\prod_{i=1}^{k}(z-z_i)^{-\frac{1}{12}}|^2$ on pg. 493 of "Supergravity vacua and solitons" by G. Gibbons.

In what other contexts in physics does the Dedekind $\eta$ function take a bow?

(Edit) Moreover, since this is a community wiki and not a test question with one best answer but an attempt to come to a better understanding of the $\eta$-function and associated math and physics, I invite people to expand on any of the items with specifics (e.g., exact formulas), more references, and/or insightful commentaries (e.g., what you believe are important aspects of the references).

Other appearances: In Gliozzi's "The Infrared Limit of QCD Effective String" on pg. 14; Panero's "A numerical study of confinement in compact QED" on pg. 4; Zahed's "Holographic Pomeron and Primordial Viscosity" on pg. 1; Caselle and Pinn's "On the Universality of Certain Non-Renormalizable Contributions in Two-Dimensional Quantum Field Theory" on pg. 3; Billo, Casselle, and Pellegrini's "New numerical results and novel effective string predictions for Wilson loops" on pg. 6 and 15; and Basar, Kharzeev, Yee, and Zahed's "Holographic Pomeron and the Schwinger Mechanism" on pg. 7.

A Dedekind Eta trajectory / horocyclic flow (Reference request)

2012-12-17T03:49:22Z

I've been exploring the composition of essentially the Dedekind $\eta$-function with parabolic Möbius transformations,

$$C_L(z,t)=\left(\frac{z}{-tz+1}\right)^{\frac{1}{2}}\eta\left(\frac{z}{-tz+1}\right)\circeq\exp(\frac{i2\pi t}{24})\: z^{\frac{1}{2}}\:\eta(z)=C_R(z,t),$$

where the symbol $\circeq$ is used to signify that equality holds only for integer $t$, and I came across the interesting parametric curves below for $z=-2+.3i$ and $-12 \leq t \leq 12$ :

$xL(t)=Real[C_L(z,t)]\:\:\:$ and $\:\:\:yL(t)=Imag[C_L(z,t)]$ and analogously for $C_R(z,t)$

$C_L(z,t)$ is annihilated by $\frac{\partial }{\partial t}-z^2\frac{\partial }{\partial z}$, while $C_R(z,t)$ is not, even at integer $t$.

Letting $z \mapsto -\frac{1}{z} $, gives $$C_L^i(z,t)=\eta(z+t)\circeq\exp(\frac{i2\pi t}{24})\: \eta(z)=C_R^i(z,t)$$

and the corresponding figure

$C_L^i(z,t)$ is annihilated by $\frac{\partial }{\partial t}-\frac{\partial }{\partial z}$, while $C_R^i(z,t)$ is not, even at integer $t$.

I've scanned through quite a lot of papers containing info on the $\eta$-function yet haven't seen similar figures, but the Dedekind $\eta$ has been pretty well explored, so I was hoping someone could direct me to some references in the vast literature that might explain the geometry of such trajectories. (Obviously, a torus is evoked, but ....)

I'm aware that E. Ghys deals with similar topics in "Knots and Dynamics" (see also Site1 and Site2), but I'm not sure (yet) how to clearly connect his arguments to the above curves.

Answer by Tom Copeland for Are there other nice math books close to the style of Tristan Needham?

2012-07-20T03:32:32Z

The Shape of Algebra in The Mirrors of Mathematics by G. Katz and V. Nodelman

and

The Wild World of 4-Manifolds by Alexandru Scorpan

Conjugation: The nature of the beast--spawn, perspectives, broadest formulation

2012-11-18T03:28:57Z

By conjugation, I mean any mathematical maneuver of the basic form

$$O=KPK^{-1}\Leftrightarrow P=K^{-1}OK.$$

What are some interesting examples of, perspectives on, and/or broad (broadest?) enlightening formulations of this relation?

(Questions about the general nature of this beast have re-surfaced most recently in a sojourn in Givental's "Gromov-Witten invariants and quantization of quadratic invariants" where there's a plethora of its spawn.)

Examples:

1) $$x^{-a}\:\frac{\widehat{Dx}^n}{n!}\:x^{a}=L^a_n(-\widehat{xD})\Leftrightarrow\frac{\widehat{Dx}^n}{n!}=x^{a}L^a_n(-\widehat{xD})x^{-a} $$

where by definition $\widehat{AB}^n=A^nB^n$, $D=d/dx$, and $L^a_n(x)$ are the associated Laguerre polynomials of order $a$ and degree $n$. This a a good analog of a matrix similarity transformation with the Laguerre polynomials represented as operators, but also, when another conjugation with $e^{x}$ is performed, reducing the ops to simple functions, they serve as an orthonormal basis for resolving functions. (OEIS-A185896 is another interesting, but less general example.)

2) $$x^{1/2}\:D^{m+1}\:x^{-1/2}=L_m\Leftrightarrow D^{m+1}=x^{-1/2}\:L_m\:x^{1/2}$$

where $D=x\:\frac{d}{dx}\: x$ (see A132440 and comments in A007318 on this classical operator and its relation to ladder ops). Here $L_m$ ($m=-1,0,1,...$) is a mutation of the ops defined on pg. 3 of Givental's paper, related to infinitesimal symplectic transformations and the infinite dimensional Lie Witt / Virasoro algebras.

3) (Path ordered exponential related to Wilson loops)

$$g(x) \mathcal{P}e^{i \oint_C A_\mu dx^\mu} g^{-1}(x)=\mathcal{P}e^{i \oint_C A_\mu dx^\mu}$$

where $\mathcal{P}$ is the path ordering operator, $A_\mu$ is gauge field, $C$ is a closed curve in space, and $g$ represents a gauge transformation. E.g., for $SU(2)$ gauges, $g^{\pm 1}(x)\equiv\exp{\pm i\alpha^j(x)\frac{\sigma^j}{2}}$, where $\alpha^j(x)$ is an arbitrary real function of $x$ and $\sigma^j$ are the three Pauli matrices.

$$tr(A)=\ln[det[\exp(A)]] \Leftrightarrow det(A)=\exp[tr[\ln(A)]]$$

See MO-Q111770 and the Cayley-Hamilton theorem for some consequences of this conjugation.

Please feel free to elaborate on these examples.

Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

2012-11-11T04:25:51Z

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?

I've found some examples:

1) In MO-Q111339 on a Tamagawa number, GH states

$$\mathrm{vol}(\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z}))=\zeta(2).$$

2) In "Quantum Gauge Theories in Two Dimensions," Edward Witten derives

$$\mathrm{vol}(\mathcal M)=\frac{2}{(\sqrt{2}\:\pi)^{2g-2}}\zeta(2g-2)$$

from a volume form for the moduli space $\mathcal M$ of flat connections on a gauge group ($G=SU(2)$) bundle over a compact two-dimensional manifold, a Riemann surface of genus $g$, and, for a connected sum of an orientable surface of genus $g$ with $k$ Klein bottles and $r$ copies of the projective plane $RP^2$, he derives

$$\mathrm{vol}(\mathcal M)=\frac{2(1-2^{1-(2g-2+2k+r)})}{(\sqrt{2}\:\pi)^{2g-2+2k+r}} \zeta(2g-2+2k+r).$$

3) In Wikipedia on the Stefan-Boltzmann law, the black body irradiance (total energy radiated per unit surface area of a black body per unit time) is given as

$$j^{*}=2\pi\:3!\zeta(4)\:\frac{(kT)^{4}}{c^{2}h^{3}}.$$

(In n-dimensional space, it's proportional to $n!\zeta(n+1)$, and Planck's law for the electromagnetic energy density inside the 3-D black body has an extra factor of $4/c$.)

4) In "Feynman's Sunshine Numbers," David Broadhurst gives the rate per unit surface area at which a black body at temperature $T$ emits photons as

$$2\pi\:2!\zeta(3)\:\frac{(kT)^{3}}{c^{2}h^{3}}.$$

(And the density of photons inside the body has an extra factor of $4/c$.)

Motivation: I'm motivated not only by general interest, but also by MO-Q111165 and MO-Q111770. Determinants (volumes) of adjacency matrices and, therefore, the cycle index polynomials (CIPs) for the symmetric group pop up in statistical physics, e.g., in Potts q-color field theory and scaling random cluster model, and the CIPS can be "rescaled" to obtain the complete Bell polynomials (OEIS-A036040) which are related to the cumulant expansion polynomials (OEIS-A127671), both of which are related to statistical correlations and their diagrammatics (see references in OEIS-A036040).

5) The $p_n(z)$ of MO-Q111165 seem formally related to the Chern classes $c_{k}(V)$ of a direct (infinite) sum of line bundles $\:\:\:\: V=L_{1}\oplus L_2\oplus ...\:.$ :

With $x_{i}=c_{1}(L_i)$, the first Chern classes,

$$p_k(z)=k!\:c_{k}(V)=k!\:e_{k}(x_{1},x_{2}, ...),$$

where $e_{k}$ are elementary symmetric polynomials. The $\zeta(n)$ can be identified as the power sums of the first Chern classes, and then, for example,

$$3!\:c_{3}(V)=p_3(z)=(z+\gamma)^3-3\zeta(2)(z+\gamma)+2\zeta(3)$$ $$4!\:c_{4}(V)=p_4(z)=(z+\gamma)^4-6\zeta(2)(z+\gamma)^2+8\zeta(3)(z+\gamma)+3[\zeta^2(2)-2\zeta(4)].$$

Update (Nov. 16, 2012): Just found the sequence in a thesis by R. Lu, "Regularized Equivariant Euler Classes and Gamma Functions," which discusses the relationship to Chern and Pontrjagin classes.

See also "An integral lift of the Gamma-genus" and "The motivic Thom isomorphism" by Jack Morava and "Hodge theoretic aspects of mirror symmetry" by L. Katzarkov, M. Kontsevich, and T. Pantev.

Answer by Tom Copeland for Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

2012-11-11T22:49:25Z

Elaborating on Nash's comment:

Oliver, special case of Zipf's law, right? That leads to the Zipf–Mandelbrot law that has a probability mass function of $$f(k;N,1,s)=\displaystyle\frac{\frac{1}{(k+1)^s}}{\sum_{i=1}^{N}\frac{1}{(i+1)^s}}$$ and then back to $\mathrm{vol}(\mathcal M)$ for the Klein bottles and particle statistics through $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^{\infty } \frac{1}{2^{n+1}} \sum_{k=0}^{n}(-1)^k \binom{n}{k}\frac{1}{(k+1)^s}$$ $$=\eta(s)=\int_{0}^{\infty }\frac{1}{\exp(x)+1}\frac{x^{s-1}}{(s-1)!}dx$$

where $\eta(s)$ is the Dirichlet eta function, and so the Klein bottle manifolds seem connected to fermions and Fermi-Dirac statistics (as apropos Möbius twists), whereas the orientable Riemann manifolds seem related to bosons and Bose-Einstein statistics.

And, Alan Gut in "Some remarks on the zeta distribution" defines the random variable $U$ with probability mass function (choose your favorite $\sigma= 2, 3, ...$)

$$P(U_\sigma)=\frac{1}{\zeta(\sigma)n^\sigma}$$

and says, "The main point is that, for $\sigma>1$, one can view the normalized zeta function $\varphi_{\sigma}(t)=\frac{\zeta(\sigma\:+\:i\:t)) }{\zeta(\sigma)}$ as the characteristic function of, as it turns out, a compound Poisson distribution. "

He shows how the moments and cumulants of the distribution (related to OEIS A036040 and A127671) given as functions of $\zeta(\sigma)$ and its derivatives are related to the von Mangoldt and Moebius functions and re-derives (and extends) an identity due to Selberg.

On a tangent, the zeta values can be used to translate the Gamma-genus:

With $$R_z = z+\gamma + \sum_{n=1}^{\infty } (-1)^n\zeta (n+1)(d/dz)^n,$$

then $$\displaystyle \exp(\omega\:R_z)\frac{e^{(t\:z)}}{t!}=\exp{(\omega\:d/dt)}\frac{e^{(t\:z)}}{t!}=\frac{e^{((t+\omega)\:z)}}{(t+\omega)!}$$

Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

2012-11-01T15:48:45Z

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of polynomials for which $\frac{d}{dx}p_n(x)=np_{n-1}(x)$, that is defined by the following recursion relation:

$p_{0}(x)=1$, $p_{1}(x)=x+\gamma$, and for $n>0$ $$p_{n+1}(x)=(x+\gamma)p_{n}(x)+\sum_{j=1}^{n}(-1)^j\binom{n}{j}j!\zeta (j+1)p_{n-j}(x)$$

where $\gamma=-\frac{\mathrm{d} }{\mathrm{d} \beta }\beta !\mid_{\beta =0 }$, the Euler-Mascheroni constant, and $\zeta(s)$ is the Riemann zeta function.

They satisfy $$p_{n}(x)=\frac{\mathrm{d^n} }{\mathrm{d} \beta^n }\frac{\exp(\beta x)}{\beta !} \mid_{\beta =0 }.$$

Explicitly,

$$p_2(x)=(x+\gamma)^2-\zeta(2)$$ $$p_3(x)=(x+\gamma)^3-3\zeta(2)(x+\gamma)+2\zeta(3)$$ $$p_4(x)=(x+\gamma)^4-6\zeta(2)(x+\gamma)^2+8\zeta(3)(x+\gamma)+3[\zeta^2(2)-2\zeta(4)]$$ $$p_5=p_1^5-10\zeta(2)p_1^3+20\zeta(3)p_1^2+15[\zeta^2(2)-2\zeta(4)]p_1+4[-5\zeta(2)\zeta(3)+6\zeta(5)]$$

Update: The coefficients appear related to OEIS-A055137, coefficients of the characteristic polynomial of the adjacency matrix of the complete n-graph.

Can anyone provide a reference for these polynomials or point out an interesting combinatorial interpretation?

Background: Rich associations with fractional calculus, digamma function, ladder operators

The fractional integro-derivative can be represented as an exponentiated convolutional infinitesimal generator (cf. MSE-Q125343):

$\displaystyle\frac{d^{-\beta}}{dx^{-\beta}}\frac{x^{\alpha}}{\alpha!}= \displaystyle\frac{x^{\alpha+\beta}}{(\alpha+\beta)!} = exp(-\beta R_x) \frac{x^{\alpha}}{\alpha!}$

where

$$R_xf(x)=\frac{1}{2\pi i}\displaystyle\oint_{|z-x|=|x|}\frac{-ln(z-x)+\lambda}{z-x}f(z)dz$$

$$=(-ln(x)+\lambda)f(x)+\displaystyle\int_{0}^{x}\frac{f\left ( x\right )-f(u)}{x-u}du.$$

with $\lambda=d\beta!/d\beta|_{\beta=0}$. (Note the integrand is related to the q (Jackson) derivative, and the Pincherle derivative / commutator is $[R_x,x]=D_x^{-1}$.)

Then $$exp(-\beta R_x) 1 =\displaystyle\frac{x^\beta}{\beta!} = exp(-\beta\psi_{.}(x)), $$

with $(\psi_{.}(x))^n=\psi_n(x)$, which implies

$$\psi_{n}(x)=(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0},$$ $$L_x\psi_{n}(x)=n\psi_{n-1}(x)=-x\displaystyle\frac{d}{dx}\psi_{n}(x),$$ $$R_x\psi_{n}(x)=\psi_{n+1}(x).$$

Let $x=e^z$ and $p_n(z)=(-1)^n \psi_{n}(e^z)$. Then

$$exp(-\beta R_z) 1 =\displaystyle\frac{exp(\beta z)}{\beta!} = exp(\beta p_{.}(z)), $$

$$L_z p_{n}(z)=n p_{n-1}(x)=\displaystyle\frac{d}{dz} p_{n}(z),$$ $$R_z p_{n}(z)= p_{n+1}(z)= (z+\gamma)p_n(z)-\displaystyle\int_{-\infty}^{z}\frac{p_n\left ( z\right )-p_n(u)}{e^z-e^u} e^u du$$

with $\gamma=-d\beta!/d\beta|_{\beta=0}$, the Euler-Mascheroni constant.

Since $p_n(z)$ is an Appell sequence and, consequently, $p_n(x+y)=(p.(x)+y)^n$, umbrally, a change of integration variables $\omega=z-u$ gives

$$R_z p_{n}(z)= p_{n+1}(z)= (z+\gamma)p_n(z)-\displaystyle\int_{0}^{\infty}[p_n(z)-(p_{.}(z)-\omega)^n] \frac{1}{e^{\omega}-1}d\omega$$

from which the recursion formula follows.

In addition, using the operator formalism for Sheffer sequences, of which the Appell is a special case,

$$R_z=z-\frac{\mathrm{d} }{\mathrm{d} \beta}ln[\beta!]\mid _{\beta=\frac{\mathrm{d} }{\mathrm{d} z}=D_z}=z-\Psi(1+D_z)$$

where $\Psi(x)$ is the digamma or Psi function.

UPDATE (Nov. 16, 2012): Just found this exact sequence in the thesis "Regularized Equivariant Euler Classes and Gamma Functions" by R. Lu with a discussion of the relationships to Chern and Pontrjagin classes.

Answer by Tom Copeland for Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

2012-11-03T11:55:23Z

Follow-up on Rupinski's and Chapoton's observations:

To nail down the identification of the $p_n(x)$ with the cycle index polynomials for $S_n$ (or the partition polynomials of the refined Stirling numbers of the first kind A036039), look at the Taylor series rep of the digamma operator for the raising / creation operator for the $p_n(z)$ basis

$$R_z = z-\Psi(1+D_z) = z+\gamma + \sum_{n=1}^{\infty } (-1)^n\zeta (n+1)D_z^n.$$

This is precisely the raising operator for the cycle index polynomials as presented on page 23 of Lagrange à la Lah Part I with $c_1=z+\gamma=p_1(x)$ and $c_{n+1}=(-1)^n\zeta(n+1)$ for $n>0$

$$D^{-1}_{c_1}= :\frac{c_{.}}{1-c_{.}D_{c_1}}: = c_1+\sum_{n=1}^{\infty } c_{n+1}D_{c_1}^n.$$

Alternatively, the Newton identities extrapolated to an entire function as an infinite order polynomial using the Weierstrass factorization maneuver can be applied to see the connections to the power and elementary symmetric polynomial formalism:

$$\exp\left (-\beta p_{.}(z)\right )=\frac{\exp\left (-\beta z \right )}{\left (-\beta \right )!}=\exp\left (-\beta(z+\gamma) \right )\prod_{k=1}^{\infty }\left ( 1-\frac{\beta}{k} \right )\exp\left (\frac{\beta}{k} \right )$$

$$=\exp\left [-(z+\gamma)\beta -\sum_{k=2}^{\infty } \frac{\zeta (k)\beta ^k}{k} \right ]=\exp\left [ :ln(1-a\beta ) :\right ]$$ where $a^1=a_{1}=(z+\gamma)$ and $a^k=a_k=\zeta(k)$ for $k>1$, but this is precisely the umbral form of the e.g.f. for the cycle index polynomials (mod signs).

(Also there are connections to rational zeta series.)

Update (Nov. 16, 2012): The generating series appears on pg. 58 in "Hodge theoretic aspects of mirror symmetry" by L. Katzarkov, M. Kontsevich, and T. Pantev (following Lu's references).

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

2012-11-08T00:52:10Z

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta function; and a special Reulle (aka dynamical systems or Smale) zeta function, the Ihara zeta function for a graph $G$--all can be expressed in the same basic form:

$$\zeta(u)=\exp\left ( \sum_{m\geq 1} \frac{N_mu^m}{m} \right ).$$

For graph zeta functions $\zeta(u,G_n)$ typically $N_m$ is the number of closed walks of $m$ steps (with some qualifications) on the graph $G$ with $n$ vertices and is related to the trace of the power of an adjacency matrix $A_n$, i.e., $N_m = tr[A_n^m]$ (e.g., A054878 and A092297).

In this case, you can use the general heuristic $O=KPK^{-1}\Leftrightarrow P=K^{-1}OK$ to obtain

$$tr(A)=\ln[det[\exp(A)]] \Leftrightarrow det(A)=\exp[tr[\ln(A)]]$$

and then

$$det(I-uA_n)=\exp[tr[ln(I-uA_n)]]=\exp\left( -\sum_{m\geq 1} \frac{tr(A_n^m)u^m}{m} \right)$$ $$=\exp\left (-\sum_{m\geq 1} \frac{N_mu^m}{m} \right ),$$

$$\zeta(u;G_n)=\frac{1}{det(I-uA_n)}=\exp\left(\sum_{m\geq 1} \frac{tr(A_n^m)u^m}{m} \right)=\exp\left(-:\ln(1-ua): \right).$$ where $a^k=a_k=tr(A_n^k)$ for $k>0$.

This last expression is the umbral form for the exponential generating function for the cycle index polynomials (OEIS-A036039) for the symmetric group (mod signs).

The Appell sequence in MO-Q111165 incorporating the Riemann zeta function reverses the last relation in some sense:

$$\exp\left (-\beta p_{.}(z)\right )=\exp\left [-(z+\gamma)\beta -\sum_{k=2}^{\infty } \frac{\zeta (k)\beta ^k}{k} \right ]=\exp\left [ :ln(1-b\beta ) :\right ]$$ where $b^1=b_{1}=(z+\gamma)$ and $b^k=b_k=\zeta(k)$ for $k>1$.

For easy reference: $$p_{0}(x)=1$$ $$p_{1}(x)=x+\gamma$$ $$p_2(x)=(x+\gamma)^2-\zeta(2)$$ $$p_3(x)=(x+\gamma)^3-3\zeta(2)(x+\gamma)+2\zeta(3)$$ $$p_4(x)=(x+\gamma)^4-6\zeta(2)(x+\gamma)^2+8\zeta(3)(x+\gamma)+3[\zeta^2(2)-2\zeta(4)]$$

These polynomials are the first few cycle index polynomials for the symmetric group. I'd like to relate each $p_n(x)$ to the characteristic polynomial of a matrix with a null main diagonal.

For example, for such a 3x3 matrix the char polynomial is

$$ \sigma^3-(a_{12}a_{21}+a_{13}a_{31}+a_{23}a_{32})\sigma+(a_{12}a_{23}a_{31}+a_{13}a_{32}a_{21}).$$

Picture a triangle with the vertices (v) labelled 1 to 3. Make a closed loop or path traversing the triangle from v_1 through v_2 and v_3 and then to v_1. Denote this closed transition/loop/path of three steps and length three by $a_{12}a_{23}a_{31}$ and assign it the "moment/transition amplitude" of $\zeta(3)$. Likewise, assign the amplitude $\zeta(2)$ to paths of two steps and length one $a_{12}a_{21}$, an amplitude of $\sigma=x+\gamma$ to a self- or null-loop, and so on. This generates $p_3(x)$.

The analogous 4x4 determinant generates six paths each with four steps and length four, e.g., $a_{12}a_{24}a_{43}a_{31}$, that can be assigned an amplitude of $\zeta(4)$ each and three sets of two paths of two steps and length one, e.g., $a_{13}a_{31}a_{24}a_{42}$, that can be assigned an amplitude of $\zeta^{2}(2)$. The algorithm can be continued to the other terms to generate $p_4(x)$.

How to prove that the algorithm will work for all $p_n(x)$, i.e., that each $p_n(x)$ can be generated in the above manner from an $n$ by $n$ "adjacency" matrix?

What properties should a transform have to deserve the descriptor Fourier?

2012-10-08T09:41:59Z

Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions:

1) What properties do you feel are essential for a transform to possess to be called a "Fourier" transform?

2) What properties of the classical Fourier transform are not necessarily shared by a generalized "Fourier" transform?

In other words, how can I recognize a "Fourier" transform?

Answer by Tom Copeland for What is Lagrange Inversion good for?

2011-01-12T11:16:27Z

The reps of the Lagrange inversion formula (LIF) in different “coordinate systems” are intrinsically interesting.

Consider a compositional inverse pair of functions, $h$ and $h^{-1}$, analytic at the origin with $h(0)=0=h^{-1}(0)$.

Then with $\omega=h(z)$ and $g(z)=1/[dh(z)/dz]$,

$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]f(z) = \exp \left[ {t\frac{d}{{d\omega }}} \right]f[{h^{ - 1}}(\omega )] = f[{h^{ - 1}}[t + \omega]] = f[{h^{ - 1}}[t + h(z)]],$$ so $$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]z |_{z=0}=h^{-1}(t)$$

(see OEIS A145271 and A139605 for more relations).

With the power series rep $h(z)= c_1z + c_2z^2 + c_3z^3 + ... ,$

$$\frac{1}{5!}[g(z)\frac{d}{{dz}}]^{5}z|_{z=0} = \frac{1}{c_1^{9}} [14 c_2^{4} - 21 c_1 c_2^2 c_3 + c_1^2[6 c_2 c_4+ 3 c_3^2] - 1 c_1^3 c_5],$$

which is the coefficient of the fifth order term of the power series for $h^{-1}(t)$. This is related to a refined f-vector (face-vector) for the 3-D Stasheff polytope, or 3-D associahedron, with 14 vertices (0-D faces), 21 edges (1-D faces), 6 pentagons (2-D faces), 3 rectangles (2-D faces), 1 3-D polytope (3-D faces).

This correspondence between the refined f-vectors of the n-Dim Stasheff polytope, or associahedron, and the coefficients of the (n+2) term of the compositional inverse holds in general, (see A133437, inversion for power series, and compare with A033282, coarse f-vectors for associahedra, and with MO-6373). These refined partition polynomials are also a refined presentation of the number of diagonal dissections of a convex n-gon (A033282) or, equivalently, the refined numbers for a set of Schroeder lattice paths (A126216), which sum to the little Schroeder numbers (A001003).

If $h(z)$ is presented as a Taylor series, the LIF A134685 is obtained, which is related to the tropical Grassmannian G(2,n) and phylogenetic trees (A134991) and to the partitioning of 2n elements into n groups.

When the invertible function $h(t)$ is represented as a power series of its own reciprocal, $t/h(t)$, the refined Narayana numbers are obtained (A134264), which are the refined h-polynomials of the simplicial complexes (A001263) dual to the Stasheff associahedra and also a refined presentation of a set of Dyck lattice paths A125181, which sum to the Catalan numbers A000108.

Also, the "infinitesimal generators" A145271 for these reps have very interesting associations (e.g., to permutahedra, surjections, and multiplicative reciprocals A019538/A049019, for the LIF A134685) and allow reps of the partition polynomials for A133437 as colored umbral binary trees related to refined Lah polynomials.

To illustrate an important application, you might look at OEIS-A074060 "Graded dimension of the cohomology ring of the moduli space of n-pointed curves of genus 0 satisfying the associativity equations of physics (also known as the WDVV equations)," as well as links in the LIF entries, to relate Lagrange inversion (or, equivalently, the Legendre transform) of series to the cohomology of moduli spaces.

For a less fancy application, the LIFs can sometimes be used, just as other transforms, such as the Fourier transform, to jump between "reciprocal" domains to simplify expressions to solve a problem, e.g., in conjunction with the OEIS to suggest generating functions for integer arrays by looking at their compositional inverses numerically.

Explaining Mukai-Fourier transforms physically

2012-05-19T01:19:28Z

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).

The basic algorithm is to find dual sets of eigenvectors/eigenfunctions parametrized by a continuous (e.g., $\omega$ below) or discrete index (e.g., $n$ below), that satisfy completeness and orthogonality relations encapsulated in Dirac delta function resolutions such as that for the FT

$$\delta(x-y)= \int_{-\infty}^{\infty}\exp(i2\pi \omega x)\exp(-i2\pi \omega y)d\omega$$

giving

$$\int_{-\infty}^{\infty}f(y)\delta(x-y)dy=f(x)=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\int_{-\infty}^{\infty}f(y)\exp(-i2\pi \omega y) dy d\omega$$

$$=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\hat{f}(\omega) d\omega,$$

or that for the eigenvectors of Sturm-Liouville differential operators over finite domains

$$\delta(x-y)=\sum_{n=0}^{\infty }\Psi_n(x)\Psi_n^*(y)$$

giving

$$f(x)=\sum_{n=0}^{\infty }\Psi_n(x)\int_{a}^{b}f(y)\Psi_n^*(y) dy,$$

or Kronecker delta resolutions such as that for the associated Laguerre functions

$$\frac{(n+\alpha)!}{n!}\delta_{mn}=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$$

giving

$$f(x)=\sum_{n=0}^{\infty }\frac{n!L_{n}^{\alpha}(x)}{(n+\alpha)!}\hat{f}_n$$

with

$$\hat{f}_n=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$

The basic "physical" operation (BPO) at work here can be regarded as destructive/constructive interference; the product at a point of the value of the function (to be resolved) with the corresponding value of an eigenfunction has a negative or positive value (or phase factor) that may sum constructively or destructively with products at other points (seen as a matched filtering or correlation by replacing $y$ with $x-z$ above). Alternatively, the BPO may be viewed as projection of vectors onto a set of orthonormal axes. In addition, if the function and operations are discretized and/or the domains restricted (in one space or its dual or both, as for the DFT) aliasing (which seems analogous to the introduction of equivalence classes) is introduced and periodicity imposed.

Can you explain the machinery behind the Mukai-Fourier transform in terms of these BPOs or close analogies?

Answer by Tom Copeland for History of the Sampling Theorem

2012-05-17T08:40:42Z

As a start to a more comprehensive search, some notes on interpolation using the Dirichlet and Fejer kernels, close cousins of the sinc kernel, can be found in a eulogy for Fejer.

And, you yourself in your answer to MO-Q58325 present a link to a paper by J. de Seguier, published in 1892, that has a Dirichlet kernel interpolation formula and a series that looks suspiciously like a sinc interpolation with the bandwidth $\omega$.

Edit: In the old days, the sinc function was referred to as the cardinal interpolation function and sinc function interpolations as cardinal series. Here is an article (1927) by J. M. Whittaker (son of E. T.:) The "Fourier" Theory of the Cardinal Function in which you can find the nascent Whittaker-Shannon sampling theorem, but E. T. Whittaker published an earlier one in 1915 as discussed by H. D. Luke in The Origins of the Sampling Theorem.

(Also of interest) A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing (paper) (website)

Answer by Tom Copeland for Pochhammer symbol of a differential, and hypergeometric polynomials

2012-09-14T15:40:57Z

Formally using the inverse Mellin transform for x>0:

$$e^x f(x\tfrac{d}{dx})e^{-x}=e^x f(x\tfrac{d}{dx}) \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} \frac{x^{-s}}{(-s)!} ds$$

$$=e^x \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} f(-s) \frac{x^{-s}}{(-s)!} ds.$$

Let $$f(x)=\binom{x+\alpha+\beta}{\beta},$$

then

$$e^x \binom{x\tfrac{d}{dx}+\alpha+\beta}{\beta}e^{-x}=L_{\beta}^{\alpha}(x)=\binom{\alpha+\beta}{\beta} K(-\beta,\alpha+1,x)$$

where $L_{\beta}^{\alpha}(x)$ is the generalized Laguerre function and $K(-\beta,\alpha+1,x),$ Kummer's confluent hypergeometric function.

For an elaboration, see the notes The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions.

See also Rodriguez-like formula on pg. 59 of Bateman's (et al.) Higher Transcendental Functions Vol. I:

$$(x\tfrac{d}{dx}+\alpha)_{n} h(x) = x^{1-\alpha}D^{n}[x^{n+\alpha-1}h(x)],$$

with $D=\tfrac{d}{dx}$, leading to

$$e^x \binom{x\tfrac{d}{dx}+\alpha+n}{n}e^{-x}=e^x x^{-\alpha}\tfrac{D^{n}}{n!}[x^{n+\alpha}e^{-x}]=L_{n}^{\alpha}(x).$$

This can be generalized by using the fractional integro-derivative representation of $K(a,b,x)$ (see Eqn. 13.2.1 on pg. 505 of Abramowitz and Stegun):

$$K(a,b,x)= e^x \tfrac{(b-1)!}{x^{b-1}}\int_{0}^{x} e^{-t}\tfrac{(x-t)^{a-1}}{(a-1)!} \tfrac{t^{b-a-1}}{(b-a-1)!} dt=e^x \tfrac{(b-1)!}{x^{b-1}}D^{-a}[e^{-x}\tfrac{x^{b-a-1}}{(b-a-1)!}],$$

leading to

$$e^x {x^{-\alpha}}\tfrac{D^{\beta}}{\beta!}[x^{\beta+\alpha}e^{-x}]=L_{\beta}^{\alpha}(x)=\binom{\alpha+\beta}{\beta} K(-\beta,\alpha+1,x).$$

Why do polytopes pop up in Lagrange inversion?

2011-10-11T00:24:37Z

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for compositional inversion.

Background update (8/2012): Consider a compositional inverse pair of functions, $h$ and $h^{-1}$, analytic at the origin with $h(0)=0=h^{-1}(0)$.

Then with $\omega=h(z)$ and $g(z)=1/[dh(z)/dz]$,

(see OEIS A145271 and A139605 for more relations).

With the power series rep $h(z)= c_1z + c_2z^2 + c_3z^3 + ... ,$

$$\frac{1}{5!}[g(z)\frac{d}{{dz}}]^{5}z|_{z=0} = \frac{1}{c_1^{9}} [14 c_2^{4} - 21 c_1 c_2^2 c_3 + c_1^2[6 c_2 c_4+ 3 c_3^2] - 1 c_1^3 c_5],$$

This correspondence between the refined f-vectors of the n-Dim Stasheff polytope, or associahedron, and the coefficients of the (n+2)-th term of the compositional inverse holds in general, (see A133437, inversion for power series, and compare with A033282, coarse f-vectors for associahedra, and with MO-6373).

(If $h(z)$ is presented as a Taylor series, the LIF A134685 is obtained, which is related to A134991 [tropical Grassmannian G(2,n)], and using the reciprocal of $h(z)$, the LIF A134264 is obtained, which is related to the Narayana triangle A001263 [h-vectors of dual of associahedra].)

Why (morally/intuitively, vague notion) do the refined face numbers of the associahedra appear as the coeficients of Lagrange inversion/reversion for a power series, or ordinary generating fct., as presented in OEIS A133437?

Loday expresses a similar interest on page 15 of "The Multiple Facets of the Associahedron" in Sec. 6 Inversion of Power Series. He ends with "There exists a short operadic proof of the above formula [LIF essentially] which explicitly involves the parenthesizings [of associahedra], but it would be interesting to find one which involves the topological structure of the associahedron."

One viewpoint, for example: I can derive the LIF several ways and relate the methods to rooted trees and thence to associahedra, but is there an intuitive way to relate the LIF for compositional inversion (which is related to the Legendre transformation/Legendre-Fenchel transform) to the geometry of the associahedra through a geometrical view of optimization via integer programming? Compositional inversion and the Legendre transformation have geometrical interpretations and are related to optimization as discussed by Strang in his book Intro. to Applied Mathematics (see also Mathemagical Forests and references therein in the section A Walk With Lagrange and Legendre). De Loera, Rambau and Leal in Triangulations of Set Points in Sec. 1.2 Optimization and Triangulations discuss connections of secondary polytopes to optimization.

Second viewpoint: Stasheff associahedra are intimately related to the moduli spaces of colliding particles (Devadoss, Devadoss/Heath/Vipismakul, Devadoss/Fehrman/Heath/Vashist). String interactions generate the moduli spaces of Riemann surfaces (Zwiebach, A First Course in String Theory, pg. 310) with punctures corresponding to particles interacting on a line segment. There is much literature on the relations among compositional inversion/Legendre transformation, Feynman functional/path/gaussian integrals representing partition functions and sums over Feynman diagrams/graphs for point particle interactions (Connes/Marcolli's "Noncommutative Geometry, Quantum Fields and Motives" pg. 51, Borcherd pg. 34, Getzler, Manin, Abdesselam, Bergstrom and Brown). Are there analogous arguments directly in terms of sums over moduli spaces for string interactions [as for the beta integral for the Veneziano amplitudes (Zwiebach, pg. 311)] that circumvent the Feynman particle/stable graph interpretations and highlight more directly the connections between compositional inverses/Legendre transforms and the face polynomials of associahedra?
(See also MOQ 22291 and make the change of variables $x=f^{-1}(y)$ in Theo's integral and maybe a Wick rotation.)

I should have stressed earlier that refined face partition polynomials characterize the LI for o.g.f.s rather than the usual coarse face polynomials and that both sets of polynomials contain the Catalan numbers only as the number of vertices for an associahedron. The coarse polynomials are not sufficient to enumerate distinct higher dimensional facets corresponding to distinct partitions of the LI, much less the Catalan numbers alone.

"Face" numbers for tropical Grassmannian G′_2,7 simplical complex ?

2011-10-02T12:38:22Z

In an excerpt of an article by Bernd Sturmfels, I found:

Theorem 5.5. The tropical Grassmannian $G^{′}_{2,n}$ is a simplical complex known as the space of phylogenetic trees.... It is denoted by $T_n$ and is defined as follows. The vertex set consists of all unordered pairs $\left \{ A,B \right \}$ where $A$ and $B$ are disjoint subsets of $\left [ n \right ]:=\left \{ 1,2, ... , n \right \}$ having cardinality at least two, and $A \cup B=\left [ n \right ]$. Such pairs are called splits. The number of splits is $2^{n−1} − n − 1$. Two splits $\left \{ A,B \right \}$ and $\left \{ A^{'},B^{'} \right \}$ are connected by an edge in the simplicial complex $T_n$ if and only if

(5) $A \sqsubseteq {A}'$ or $A \sqsubseteq {B}'$ or $B \sqsubseteq {A}'$ or $B \sqsubseteq {B}'$.

We define $T_n$ as the largest simplicial complex having this edge graph.... In the language of algebraic combinatorics, $T_n$ is the flag complex of the compatibility graph specified by (5) on the set of all $2^{n−1} − n − 1$ splits.

Example 5.6. ($n = 6$) The two-dimensional simplicial complex $T_6$ has $25$ vertices, $105$ edges and $105$ triangles...

Question: Are $56, 490, 1260, 945$ the "face" numbers for $T_7$?

In "splendid isolation"

2012-05-20T23:59:29Z

While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. D. Luke in The Origins of the Sampling Theorem:

However, this history also reveals a process which is often apparent in theoretical problems in technology or physics: first the practicians put forward a rule of thumb, then the theoreticians develop the general solution, and finally someone discovers that the mathematicians have long since solved the mathematical problem which it contains, but in "splendid isolation."

Other interesting examples?

(Matrices and Bohr's Quantum Mechanics of course. Someone could elaborate on the sampling theorem if they wish. )

Klein's Protocols: A window into our mathematical past

2012-06-12T07:06:19Z

Klein's Protocols in over 8,000 pages recording seminars organized from 1872 to 1913 by Felix Klein and given by Klein, his colleagues, students and other invited speakers, including luminaries such as Hilbert and Minkowski, provide a unique window into our mathematical past. Eugene Chislenko and Yuri Tschinkel have presented a beautiful introduction to the corpus, noting how the breadth of the topics reflect the broad interests and knowledge of Klein (the authors claim that Klein was one of the last three grand mathemagicians able to soar over the full realm of the mathematics of their times, the other two being Hilbert and Poincare).

Klein 's Protocols is huge and handwritten in German, so I thought it would be helpful for aspiring and established mathematicians with an interest in the history of ideas if a listing were available on Math Overflow of some gems in the corpus. Can you make a contribution?

Please provide keywords and page references for any entry. E.g., personally, I would love to have a copy of the figure titled "On regular solids in 4-dimensional space" by W. I. Stringham in Vol. II on pg. 59 (Monday, November 29, 1880).

Answer by Tom Copeland for In "splendid isolation"

2012-05-28T19:47:44Z

Quantum mechanics of Born, Heisenberg, and Jordan.

From Physics in my Generation (Springer, 1969) by Max Born:

"In Gottingen we also took part in the attempts to distill the unknown mechanics of the atom out of the experimental results ... The art of guessing correct formulas ... was brought to considerable perfection ...

This period was brought to a sudden end by Heisenberg ... He cut the Gordian knot ... he demanded that the theory should be built up by means of quadratic arrays ... one must find a rule ... for the multiplication of such arrays ...

By consideration of known examples discovered by guesswork, Heisenberg found this rule ...

Heisenberg's rule left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory that I had learned from my teacher, Rosanes, in Breslau. Such quadratic arrays are quite familiar to mathematicians, and are called matrices ...

[Born writes down the now iconic qp-pq=iħ.]

My excitement over this result was like that of the mariner who, after long voyaging, sees the land from afar..."

Answer by Tom Copeland for How does one motivate the analytic continuation of the Riemann zeta function?

2012-05-19T15:18:27Z

Riemann's analytic continuations and derivation of the functional equations for $\zeta$ and $\xi$ seem quite natural and intuitive from the perspective of basic complex analysis.

Riemann in the second equation of his classic paper On the Number of Prime Numbers less than a Given Quantity (1859) writes down the Laplace (1749-1827) transform

$$\int_{0}^{+\infty}e^{-nx}x^{s-1}dx=\frac{(s-1)!}{n^s},$$ valid for $real(s)>0.$

With $n=1$ this is the iconic Euler (1707-1783) integral representation of the gamma function, and noting that

$(s-1)!=\frac{\pi}{sin(\pi s)}\frac{1}{(-s)!}$ from the symmetric relation $\frac{sin(\pi s)}{\pi s}=\frac{1}{s!(-s)!},$

this can be rewritten as

$$\frac{sin(\pi s)}{\pi}\int_{0}^{\infty}e^{-x}x^{s-1}dx=\frac{1}{(-s)!},$$

suggesting quite naturally to someone as familiar with analytic continuation as Riemann that

$$\frac{-1}{2\pi i}\int_{+\infty}^{+\infty}e^{-x}(-x)^{s-1}dx=\frac{1}{(-s)!},$$

valid for all $s$, where the line integral is blown up around the positive real axis into the complex plane to sandwich it with a branch cut for $x>0$ and to loop the origin in the positive sense from positive infinity to positive infinity. Deflating the contour back to the real axis introduces a $-exp(i\pi s)+exp(-i\pi s)=-2isin(\pi s)$. (This special contour is now called the Hankel contour after Hermann Hankel (1839-1873), who became a student of Riemann in 1860 and published this integral for the reciprocal gamma fct. in his habilitation of 1863. Most likely Riemann introduced him to this maneuver.)

Riemann in his third equation observes that

$$(s-1)!\zeta(s)=(s-1)!\sum_{n=1}^{\infty }\frac{1}{n^s}=\int_{0}^{+\infty}\sum_{n=1}^{\infty }e^{-nx}x^{s-1}dx=\int_{0}^{+\infty}\frac{1}{e^x-1}x^{s-1}dx$$

and then immediately writes down as his fourth equality the analytic continuation

$$2sin(\pi s)(s-1)!\zeta(s)=i\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x-1}dx,$$

valid for all $s$, which can be rewritten as

$$\frac{\zeta(s)}{(-s)!}=\frac{-1}{2\pi i}\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x-1}dx$$

as naturally suggested by the analytic continuation of the reciprocal of gamma above.

For $m=0,1,2, ...,$ this gives

$$\zeta(-m)=\frac{(-1)^{m}}{2\pi i}\oint_{|z|=1}\frac{m!}{z^{m+1}}\frac{1}{e^z-1}dz=\frac{(-1)^{m}}{m+1}\frac{1}{2\pi i}\oint_{|z|=1}\frac{(m+1)!}{z^{m+2}}\frac{z}{e^z-1}dz$$

from which you can see, if you are familiar with the exponential generating fct. (e.g.f.) for the Bernoulli numbers, that the integral vanishes for even $m$. Euler published the e.g.f. in 1740 (MSE-Q144436), and Riemann certainly was familiar with these numbers and states that his fourth equality implies the vanishing of $\zeta(s)$ for $m$ even (but gives no explicit proof). He certainly was also aware of Euler's heuristic functional eqn. for integer values of the zeta fct., and Edwards in Riemann's Zeta Function (pg. 12, Dover ed.) even speculates that ".. it may well have been this problem of deriving (2) [Euler's formula for $\zeta(2n)$ for positive $n$] anew which led Riemann to the discovery of the functional equation ...."

Riemann then proceeds to derive the functional eqn. for zeta from his equality by using the singularities of $\frac{1}{e^z-1}$ to obtain basically

$$\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\tfrac12\pi s)\zeta(1-s),$$

and says three lines later essentially that it may be expressed symmetrically about $s=1/2$ as

$$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)=\xi(1-s).$$

Riemann then says, "This property of the function [$\xi(s)=\xi(1-s)$] induced me to introduce, in place of $(s-1)!$, the integral $(s/2-1)!$ into the general term of the series $\sum \frac{1}{n^s}$, whereby one obtains a very convenient expression for the function $\zeta(s)$." And then he proceeds to introduce what Edwards calls a second proof of the functional eqn. using the Jacobi theta function.

Edwards wonders:

"Since the second proof renders the first proof wholly unnecessary, one may ask why Riemann included the first proof at all. Perhaps the first proof shows the argument by which he originally discovered the functional equation or perhaps it exhibits some properties which were important in his understanding of it."

I wonder whether, as his ideas evolved before he wrote the paper, he first constructed $\xi(s)$ by noticing that multiplying $\zeta(s)$ by $\Gamma(\frac{s}{2})$ introduces a simple pole at $s=0$ thereby reflecting the pole of $\zeta(s)$ at $s=1$ through the line $s=1/2$ and that the other simple poles of $\Gamma(\frac{s}{2})$ are removed by the zeros on the real line of the zeta function. The $\pi^{-s/2}$ can easily be determined as a normalization by an entire function $c^s$ where $c$ is a constant, using the complex conjugate symmetry of the gamma and zeta fct. about the real axis. Riemann had fine physical intuition and would have thought holistically in terms of the the zeros of a function (see Euler's proof of the Basel problem) and its poles, the importance of which he certainly stressed.

Let's extend the reasoning above for the Jacobi theta function

$$\vartheta (0;ix^2)=\sum_{n=-\infty,}^{\infty }exp(-\pi n^{2}x^2).$$

Viewing a modified Mellin transform as an interpolation of Taylor series coefficients (MO-Q79868), it's easy to guess (note the zeros of the coefficients) that

$$\int^{\infty}_{0}\exp(-x^2)\frac{x^{s-1}}{(s-1)!} dx = \cos(\pi\frac{ s}{2})\frac{(-s)!}{(-\frac{s}{2})!} = \frac{1}{2}\frac{(\frac{s}{2}-1)!}{(s-1)!},$$

and, therefore,

$$\int^{\infty}_{0}\exp(-\pi (n x)^2)x^{s-1} dx = \frac{1}{2}\pi^{-s/2}(\frac{s}{2}-1)! \frac{1}{n^s}.$$

By now you should be able to complete the line of reasoning to obtain, for $real(s)>1,$

$$\xi(s)=\int_{0^+}^{\infty }[\vartheta (0;ix^2)-1)]x^{s-1}dx=\pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s).$$

Do an analytic continuation as done for the gamma function in MSE-Q13956 to obtain, for 0<real(s)<1,

$$\xi(s)=\int_{0^+}^{\infty }[\vartheta (0;ix^2)-(1+\frac{1}{x})]x^{s-1}dx.$$

Then use symmetries of the Mellin transform and the fact that $\xi(s)=\xi(1-s)$ (as explained in MSE-Q28737) to obtain the functional equation

$$\vartheta (0;ix^2)=\frac{1}{x}\vartheta (0;\frac{i}{x^{2}}).$$

Answer by Tom Copeland for effective teaching

2012-05-15T23:47:47Z

This very stimulating presentation on teaching math effectively came out in May 2010 and addresses some of the issues presented in the other answers:

Dan Meyer: Math class needs a makeover

Although applied to high school math, at least some aspects of the technique could be incorporated, if only in a few sessions, into advanced classes. Comments to the video by educators and students provide some feedback on the technique.

Another potential method for revamping math classes for the 21'st century (maybe start viewing at time stamp 6:50):

Salman Khan: Let's use video to reinvent education

Answer by Tom Copeland for What does Mellin inversion "really mean"?

2011-11-03T13:38:07Z

Two equations that encapsulate the properties of the Fourier and Mellin transforms:

$$\int^{\infty}_{-\infty}{\exp(2 \pi ifx)\exp(-2 \pi ify)df} = \delta(x-y)$$

$$\frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} x^{-s} y^{s} ds= \delta(\ln(x)-\ln(y))= y \delta(x-y).$$

The transformations from one equation to the other are obvious. The delta function results are intuitive and an extrapolation of the discrete case for the orthogonality relationships of the characters of character groups. The transform pairs, Plancherel and convolution theorems, and other relations are easy to derive from these two.

(Note that whereas $e^{sz}$ is an eigenfunction for $d/dz$ and so the Laplace/Fourier transforms are appropriate for devising an operator calculus for $f(d/dz)$, $z^s$ is an eigenfunction of $zd/dz$ and so the Mellin transform is more appropriate for $f(zd/dz)$.)

Ramanujan's Master Formula/Theorem (see Wikipedia, particularly the Hardy refer.) gives a somewhat intuitive perspective on the Mellin transform as providing an "interpolation" of the coefficients of the Taylor series of certain classes of functions, as discussed in the intro of "Ramanujan's Master Theorem ..." by Olafsson and Pasquale. E.g.,

$$\int^{\infty}_{0}f(x)\frac{x^{s-1}}{(s-1)!} dx = g(-s)$$ and

$$\frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} g(-s) \frac{x^{-s}}{(-s)!} ds = \sum_{n=0}^{\infty} g(n) \frac{(-x)^{n}}{n!} = f(x)$$

for the transform pairs

$f(x)=\exp(-x)$ and $g(-s)= 1$ $(\sigma>0)$ and

$f(x)=\frac{1}{1+x}$ and $g(-s)= (-s)!$ $(0<\sigma<1$ and $abs(x)<1)$

$f(x)=\exp(-x^2)$ and $g(-s)= \cos(\pi\frac{ s}{2})\frac{(-s)!}{(-\frac{s}{2})!} = \frac{1}{2}\frac{(\frac{s}{2}-1)!}{(s-1)!} $ $(\sigma>0)$.

(Note the appearance again of the Dirac delta fct. and its derivatives as $\frac{x^{-n-1}}{(-n-1)!}$.)

A simple way to derive the formulas in your question is by looking at the inverse Mellin transform rep of the Dirac delta function. See my short note on the Inverse Mellin Transform and the Dirac Delta Function. See also some applications in Dirac's Delta Function and Riemann's Jump Function J(x) for the Primes and The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions.

Edwards in Riemann's Zeta Function in Ch. 10 Fourier Analysis Sec. 10.1 Invariant Operators on R+ and Their Transforms gives a nice, more group-theoretic intro to the Mellin transform in line with other comments in this stream.

Answer by Tom Copeland for Rationale for Hadamard's finite part of a divergent integral

2012-05-04T22:19:26Z

Looking at Fourier transforms can provide an intuitive context for the Hadamard finite part (F.P.) regularization.

Monkey around with this ladder of expressions (understood as F.P.s):

$$A)\int_{-\infty }^{\infty }\frac{exp(i2\pi fx)}{(i2\pi f)^2}df=\frac{sgn(x)}{2}x$$

$$B) \int_{-\infty }^{\infty }\frac{exp(i2\pi fx)}{i2\pi f}df=\frac{sgn(x)}{2}$$ $$C)\int_{-\infty }^{\infty }exp(i2\pi fx)df=\delta(x)$$

To descend the ladder, formally take the derivative of both sides above or of the explicit F.P. expressions below (second equalities), which is equivalent to multiplying the integrands above by $i2\pi f$. To climb, integrate from $0$ to $x$ both sides below, using the explicit expressions for the integrands for the F.P. given below in the second equalities, or simply divide the integrands on the L.H.S. above by $i2\pi f$. (Note that $x$ can be negative or positive and that the Dirac delta function contributes only a value of $1/2$ when evaluated on the boundary of the integral.) So, the explicit F.P. integrals below commute with differentiation and integration w.r.t. $x$ and can be naturally defined in terms of the two ops, and the implicit symbolic formulas above allow us to formally retain the representation of the two ops as multiplication and division operations in the Fourier transform integrands.

For finite limits for the integrals, you'll end up with the expressions on the right above being convolved with a sinc function with some phase, that should agree with the L.H.S. if the Hadamard finite finesse is applied.

The OP's example is closely related to A) with $x=0$ and is more palatable within this context. In detail, in the limits $\varepsilon \to 0^+$ and $L \to \infty,$

$C)\displaystyle\delta(x)=\int_{-L }^{L }exp(i2\pi fx)df$

$B)\displaystyle\frac{sgn(x)}{2}=F.P.\int_{-\infty }^{\infty }\frac{exp(i2\pi fx)}{i2\pi f}df=\left [ \int_{\varepsilon}^{L }+\int_{-L }^{-\varepsilon} \right ]\frac{exp(i2\pi fx)-1}{i2\pi f}df$

$=\displaystyle\left [ \int_{\varepsilon}^{L }+\int_{-L }^{-\varepsilon} \right ]\frac{exp(i2\pi fx)}{i2\pi f}df-\frac{ln(L/\varepsilon)}{i2\pi}-\frac{ln(\varepsilon/L)}{i2\pi}$

$=\displaystyle\left [ \int_{\varepsilon}^{L }+\int_{-L }^{-\varepsilon} \right ]\frac{exp(i2\pi fx)}{i2\pi f}df=C.P.V\int_{-\infty }^{\infty }\frac{exp(i2\pi fx)}{i2\pi f}df$

where $F.P.$ denotes the Hadamard finite part and $C.P.V.$, the Cauchy principle value. (Of course, the $\frac{1}{f}$ terms pose no serious problems since $\frac{1}{f}$ is an odd function and we are integrating symmetrically about $0$.)

Similarly,

$A)\displaystyle\frac{sgn(x)}{2}x=F.P.\int_{-\infty }^{\infty }\frac{exp(i2\pi fx)}{(i2\pi f)^2}df=\left [ \int_{\varepsilon}^{L }+\int_{-L }^{-\varepsilon} \right ]\frac{exp(i2\pi fx)-(1+i2\pi fx)}{(i2\pi f)^2}df$

$=\displaystyle\left [ \int_{\varepsilon}^{L }+\int_{-L }^{-\varepsilon} \right ]\frac{exp(i2\pi fx)}{(i2\pi f)^2}df-\frac{2}{(i2\pi)^2 \varepsilon}=\frac{|x|}{2}.$

It's even more convincing when you plot the integrals (including C) and observe how they evolve as $L$ increases for small $\varepsilon.$

Another context for the Hadamard finite limit is given in MSE-Q13956.

For a comparison of different methods of regularization for integrals of this type see http://arxiv.org/abs/hep-th/0202023 "Improved Epstein-Glaser renormalization in coordinate space I. Euclidean framework" by Gracia-Bondia (pg. 14-).

Answer by Tom Copeland for General Procedure for Inverse of an Integral Transform

2012-04-11T02:27:16Z

Just to direct you towards the literature and help you clear up some of your thinking on the subject take a look at Wiki's Inverse Problem. Then perhaps look at Methods of Applied Mathematics by Hildebrand and Principles and Techniques of Applied Mathematics by Friedman.

Max/min problems related to associahedra or their duals (ions on balls revisited)

2012-03-14T12:22:32Z

Original motivation: This is a follow-up question to and generalization of MO Q78877 on equilibrium configurations of ions on n-Dim balls. Henry Cohn gave an excellent answer dispelling my naive intuition/hope that 14 ions would configure into the vertices of a Stasheff associahedron on a 3-Dim ball under the influence of a Coulomb potential (Thomson problem). As he remarks, the ions would configure into the vertices of deltahedra (with simplicial/triangular facets) on the 3-D ball. However, on a web page by Maurice Starck, I just noticed that a convex deltahedron with 9 vertices has 21 edges and 14 faces-the dual polyhedron to the 3-D associahedron! The 2-D case, the self-dual pentagon, is analogous. Is there a 4-D analog, i.e., does the 4-D dual polytope with 14 ions at its vertices satisfy a 4-D Thompson-like problem?

Prompted by JC's reply, I'd really like to know more generally of any (natural/enlightening) max/min problems with solutions involving the associahedra or their dual polytopes.

Answer by Tom Copeland for Why does the Laplace Transform work?

2012-03-08T00:14:03Z

Consider these relations for the Fourier, Mellin, and Laplace transforms:

$\int^{\infty}_{-\infty}{exp(2 \pi ifx)exp(-2 \pi ify)df} = \delta(x-y)$

$\frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} x^{-s} y^{s} ds= \delta(ln(x)-ln(y))= y \delta(x-y)$

$\frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} e^{-xp} e^{yp} dp=\delta(x-y)$.

For me, the delta fct. results seem intuitive (also analogous to the orthogonality relationship for characters of character groups), and the eqns. encapsulate the properties of the transforms (try deriving the transform pairs and other relations from them, e.g., Plancherel, convolution, Poisson summation) and illustrate the transformations from one transform to another.

(Tried this as a comment initially, but had formatting problems.)

Answer by Tom Copeland for Geometric picture of invariant differential of an elliptic curve

2011-12-04T11:18:47Z

A paper by John Tate (pg. 1 and 2) gives a clear derivation of the diff. form:

Reparametrize the elliptic curve

$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4 x+a_6$

with $p(z)=x+(a_1^2+4a_2)/12$ and $p^{'}(z)=2y+a_1x+a_3$ to obtain

$(p^{'})^2=4p^3-g_2p-g_3$, defining the Weierstrass elliptic fct., and

$\omega=dp(z)/p^{'}(z)=dz=dx/(2y+a_1x+a_3)$.

Per Dan's comment, a coordinate transformation of $x=u^2x^{'}+r$ and $y=u^3y^{'}+su^2x^{'}+t$
leaves $\omega^'=u\omega$.

Given $\sigma=p(z)$ and the inverse $z=p^{-1}(\sigma)$,

$dz=(p^{-1}(\sigma))^{'}d\sigma=(p^{-1}(\sigma))^{'}p^{'}(z)dz$, so

$(p^{-1}(\sigma))^{'}=1/p^{'}(z)$ and $dz=d\sigma/p^{'}(z)=\omega$.

The amplitwist interpretation of differentiation and inversion presented by Tristan Needham in his book Visual Complex Analysis provides a geometric interpretation of these differential relations.

Consider as an analogy $P(\theta)=sin(\theta), P^{'}(\theta)=cos(\theta), and P^2+(P^{'})^2=1$.

Geometric picture of invariant differential of an elliptic curve

2011-12-04T04:12:15Z

What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?

This question is an adjunct to MO Q1 on formal laws and L-series, which motivated Q2. Q1 (Silverman) and Darmon (pg. 6) state:

The invariant holomorphic differential form (Neron differential) attached to an elliptic curve is

$\omega=dx/(2y+a_1x+a_3)$.

(Ancilliary question: Relation to Weierstrass's elliptic functions?)

I'd like to broaden the question as a community wiki to ask, "What are some interesting manifestations of this one-form in various families of elliptic curves?"

E.g., J. Hoffman in Topics in Elliptic Curves and Modular Forms gives for the Jacobi quartic family of elliptic curves

$\omega=dx/(1+2\kappa x^{2}+x^{4})^{1/2}=\sum_{n=0}^{\infty}L_{n}(\kappa)x^{2n}dx$

with $L_{n}(\kappa)$ the Legendre polynomials.

Conjecture: "Neretin polynomials" for a normalized Schwarzian have integer coefficients

2012-01-17T01:17:50Z

I speculated in 2008 that the modified Neretin polynomials presented in A145900 of the On-line Encyclopedia of Integer Sequences, which can be summed to give a normalized Schwarzian derivative for a complex function and are related to a representation of the Virasoro algebra, all have integer coefficients. Definitions, references, and links are provided in the entry. Can anyone prove or disprove this conjecture?

Answer by Tom Copeland for A product approximation to the Taylor series of the exponential

2011-12-02T16:31:40Z

See A019538, A049019, and A133314 for extensive formula for the coefficients and relation to face vectors of permutahedra/permutohedra. Your proof is a nice one in that A133314 clearly shows why the lower order coefficients as given by the finite differences must vanish if $1/f_n(x)$ truly is an approximation of $exp(-x)$.

Comment by Tom Copeland

2013-04-16T10:11:09Z

@Liviu, anticipating? Check the dates. Dirac was famous for his terseness, originality, and ingenuity, as reflected in his presentation.

Comment by Tom Copeland

2013-04-16T09:42:23Z

@Terry, given the earlier stated answers below, your comment seems at best redundant. Maybe you could expand on it as an answer and provide some new details for those not already familiar with the theory of distributions

Comment by Tom Copeland

2013-04-15T23:49:57Z

Really, weren't the results for the limiting case of the derivative of the log worked out fairly rigorously by Cauchy and Poisson in their work on potential theory long before 20'th century mathematicians put a formal dress on them?

Comment by Tom Copeland

2013-04-15T22:44:22Z

Dirac would have known all these results from electrostatics to corroborate his assertion.

Comment by Tom Copeland

2013-04-15T22:39:32Z

Terry, you're missing a derivative. d/dz log(z)=1/z then look at limits as in the Poisson kernel.

Comment by Tom Copeland

2013-04-15T21:24:44Z

Dirac's bachelor's degree was in electrical engineering at a British university, so I'm sure he was familiar with and influenced by Heaviside's style of mathematics.

Comment by Tom Copeland

2013-04-15T21:05:05Z

For those who may be a little confused about the absolute value sign, just switch to polar coordinates and restrict to the real line.

Comment by Tom Copeland

2013-04-15T10:27:04Z

I'm sure Dirac was thinking that ln(x)=ln|x|+i H(-x)π, where H(x) is the Heavside step function.

Comment by Tom Copeland

2013-01-16T09:10:10Z

@Muon : Actually, you mean $z^{n+1}$. I'll expand on my comment later. I think a good answer to your questions would involve combining ideas in Khesin and Wendt's The Geometry of Infinite-Dimensional Groups, Ovsienko and Tabachnikov's Projective Differential Geometry Old and New, and the book I cited earlier in a tight narrative, but that's beyond me at the moment.

Comment by Tom Copeland

2013-01-09T23:39:02Z

@Eremenko : the paper you cite in the edit is precisely the one in my answer. Why the repetition? Luke in the cited paper states, "The first scientist to formulate the sampling theorem precisely and apply it to problems of communication engineering is probably V. A. Kotelnikov." (1933)

Comment by Tom Copeland

2013-01-09T23:12:11Z

For a physicist with an interest in QED perhaps a good place to start is Sec. 3.3 Fusion, Splitting, and Hopf Algebras in E. Zeidler's book Quantum Field Theory II Quantum Electrodynamics.

Comment by Tom Copeland

2013-01-07T05:18:04Z

Don't confuse the son J. M. (1929, and later in his book in 1935) with the father E.T. (1915) who is actually given the majority of the credit for a bandwidth analysis. (I guess you see what you look for.) If you're concerned about accreditation to Russians, you might want to confirm (or not) the chronology at the website linked to in my answer which notes Kotelnikov (1930), as well as other nationalities (Ogura, 1920).

Comment by Tom Copeland

2012-12-27T04:10:25Z

As a first step at understanding the relations, I always recall that $exp(-c \cdot z^2d/dz)f(z)=exp[c \cdot d/d(1/z)]f(1/(1/z))=f(1/(c+1/z))=f(z/(cz+1)),$ a special conformal transformation. Note $zd/dz=d/d(ln(z))$ for the dilatation.

Comment by Tom Copeland

2012-12-27T03:55:54Z

Try Ch. 9: Conformal Invariance Sec. 9.1: Energy momentum tensor-Virasoro algebra of the book Statistical Field Theory Vol. 2 by Itzykson and Drouffe. Maybe you could work around pg. 514 on translations, complex dilatations, and special conformal transformations related to $d/dz, zd/dz,$ and $z^2d/dz$.The central charge is discussed in the next sub-section from a physical and mathematical perspective.

Comment by Tom Copeland

2012-12-21T00:37:56Z

I wonder if the first curve could be lifted off the plane onto a double torus with one torus nested inside the other but sharing its circle of max radius with that of the other torus.