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It seems your conjecture is true. Mathematica gives the result $$ (1 + 4^n (-1 + n) n \mbox{Beta} [1/2, -1 + n, 2 + n] - 4^n n (1 + n) \mbox{Beta} [1/2, 1 + n, n])/(2 (1 + n)) $$ in terms of the inc …

$$ \left( \begin{array}{c} 1 \\ \frac{k_x - ik_y }{\sqrt{k_x^2 + k_y^2 } } \end{array} \right) e^{i(k_x x + k_y y)} \ \ \ \mbox{with eigenvalue} \ \ \ i\sqrt{k_x^2 + k_y^2 } $$ and $$ \left( \begin{ar …

According to the Table Errata reported in Mathematics of Computation, vol. 65, no. 215, 1996, pp. 1379–1386, this entry in Erdélyi's Tables of Integral Transforms is flawed. The exponent in the denomi …

We can obtain power series in $a$ by expanding in geometric series. Since the specific treatment depends on the integration region, let's introduce lower and upper limits, $y_0 <y$, $$ I = \int_{y_0 } …

The equation has solutions with powers, $f(x) = ax^b$. Inserting this ansatz, one has $$ a b (b-1) x^{b-2} = a (a x^b)^b = a^{b+1} x^{b^2} \ , $$ so the requirements on $a$ and $b$ are $$ b-2 = b^2 \ …

The relationship can be written using the dilation operator as $$ _2 F_1 (a,b,c,p\cdot z) = \exp \left( \ln p \cdot z \frac{\partial}{\partial z} \right)\, {}_2 F_1 (a,b,c,z) $$

There is a generalization of the variational method already shown in Linus Pauling's "Introduction to Quantum Mechanics" that isn't mentioned much anymore (possibly because of limited practical use): …

I found B. Felsager's "Geometry, Particles and Fields" a useful introduction. It has a part I that presents field theory in the language most familiar in physics, and then a part II that puts things i …

I asked Mathematica about the boundary behavior. First, the $P^{1/4}_{\nu } $ solution: We have, for $\epsilon \searrow 0$, $$ (\sin \epsilon )^{1/4} P^{1/4}_{\nu } (\cos \epsilon ) = \frac{2^{1/4} }{ …

The answer is no. A counterexample is $$ f(t,x) = x - \frac{1}{2} -\frac{1}{24} t^2 $$ $$ B(t,s,x) = \left( x-\frac{1}{2} \right) (t-s) $$ (Method: I obtained this by expanding $f$ and $B$ into power …