# Questions tagged [divergent-integrals]

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The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon. Let us consider a toroidal helix parametrized as follows: $$x=(R+r\cos(n\phi))\cos(\phi)... • 235 6 votes 3 answers 639 views ### How do I solve the following definite integral (preferably by an asymptotic method)?$$ \int_{1}^{\infty} \frac{\sin^2 (\mu \sqrt{x^2 -1})}{(x+1)^{\frac{9}{2}} (x-1)^{\frac{3}{2}}} \,dx $$Note: \mu here is an extremely small constant. I have tried: Estimating the integral by ... • 69 1 vote 0 answers 94 views ### Crazy conjecture about Bernoulli umbra and reference request For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions. Yet, it still remains mistery what ... • 9,038 2 votes 2 answers 164 views ### Assigning values to divergent oscillating integrals I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ... • 1,550 3 votes 0 answers 325 views ### Extending reals with logarithm of zero: properties and reference request If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ... • 9,038 0 votes 0 answers 14 views ### Expression for agent kinematics diverges as angular acceleration goes to 0? Given an agent on a 2D Cartesian grid initially at coordinates (0,0) pointing forwards along the x-axis ie heading (\theta) = 0, imagine that it has some initial ... -2 votes 1 answer 98 views ### Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones? [closed] Below, we interpret divergent integrals as germs of partial integrals at infinity:$$\int_0^\infty f(x) dx=\operatorname{bigpart} \int_0^\omega f(x) dx$$where \operatorname{bigpart} means taking ... • 9,038 1 vote 0 answers 93 views ### Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers? Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers B_n) as B_-, B_-+1 as B_+ (an umbra with moments being Bernoulli numbers except B_1=1/2). I will denote the ... • 9,038 1 vote 0 answers 127 views ### Do the equalities \int_0^∞1dx·\int _0^∞1dx=2\int_0^∞xdx and \int_0^∞e^xdx·\int_0^∞e^xdx=2\int_0^∞e^{2 x}dx-2\int_0^∞e^xdx make sense? Previously I tried to define multiplication of divergent integrals, but my approach turned out to be umbral-like. Now, I decided to define multiplication of divergent integrals in a Hardy fields-like ... • 9,038 0 votes 0 answers 69 views ### Generalization of Levi-Civita type construction towards divergent integrals and corresponding questions A known generalization of Levi-Civita field is a field of Hahn power series of \varepsilon of the form \mathbb{R}[[\varepsilon^{\mathbb{Q}}]]. Assuming \varepsilon=1/\omega, we can naturally ... • 9,038 1 vote 0 answers 184 views ### What's the regularized value of these divergent integrals: \int_0^\infty \ln x \, dx and \int_0^\infty \frac{\ln x}{x^2} \, dx? When playing with divergent integrals \int_0^\infty f(x) \, dx and their transformations with operators \int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx and \int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)... • 9,038 1 vote 0 answers 65 views ### There is a ring with multiplication. Can we find a formula for division based on formula for multiplication? Studying divergent integrals, I found a good formula for their multiplication: \int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)... • 9,038 6 votes 1 answer 672 views ### An operation is defined on polynomials. How do I generalize it to other classes of functions? I am currently researching divergent integrals. Definition. An extended number is an expression of the form \int_a^b f(x)\,dx, where a,b\in \overline{\mathbb{R}} and function f(x) is defined ... • 9,038 2 votes 0 answers 230 views ### Hypermodulus and what mathematical objects have it When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ... • 9,038 0 votes 0 answers 247 views ### How is this expression for the regularization of integrals of monomials, given in a paper, justified? How strong is argument in favor? In this answer by Carlo Beenakker he cites the following regularization formula:$$\int_0^\infty x^p\,dx\mathrel{"="}\frac{(-1)^{p+1}}{(p+1)(p+2)},\;\;p=0,1,2,\dotsc,$$citing Tafazoli - Calculation ... • 9,038 2 votes 1 answer 392 views ### Where do these divergent integrals appear in mathematics and physics? I have already asked a similar question, albeit far more extensive, but it was criticized and closed for being too extensive and promotional. So, here is a greatly truncated and focused version. Since ... • 9,038 1 vote 0 answers 91 views ### What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have? I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ... • 9,038 2 votes 0 answers 211 views ### Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"? There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ... • 9,038 2 votes 0 answers 147 views ### Regularization of the area under hyperbola So, I am trying to find the regularized value of the divergent integral I=\int_1^\infty \sqrt{x^2-1}dx. Since the area of \int_0^1 \sqrt{1-x^2}dx=\frac\pi4, I wonder whether the area under ... • 9,038 0 votes 1 answer 409 views ### A set of divergent integrals that I think, equal to -\gamma So, we take \frac{\text{sgn}(x-1)}{x} and apply \mathcal{L}_t[t f(t)](x) four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-... • 9,038 1 vote 0 answers 59 views ### Fractional power of the operator \mathcal{L}_t[t f(t)](x) and equivalence of divergent integrals I wonder whether an expression for fractional power of operator \mathcal{L}_t[t f(t)](x) that involves Laplace transform can be derived? I am asking this because this operator preserves the area ... • 9,038 1 vote 2 answers 252 views ### Can we meaningfully ascribe values to these divergent integrals? My gut feeling is that \int_0^\infty (1-\frac1{x^2})dx=0 \int_0^\infty (x-\frac2{x^3})dx=0 \int_0^\infty (x^2-\frac6{x^4})dx=0, etc, and in general, \int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0, ... • 9,038 3 votes 0 answers 406 views ### What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed] I am working on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities.$$\...
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I was trying to solve the following integral: $$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)}$$ The singular structure in the $z$ ...