# Tagged Questions

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

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### Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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### Integral of incomplete gamma function and limit of hypergeometric function

Let $a > 0$ and consider the integral $$\int_x^\infty \frac{\Gamma(a,t)}{t}\,dt$$ where $\Gamma(a,t)$ is the upper incomplete gamma function $$\Gamma(a,t) = \int_t^\infty x^{a-1} e^{-x} \, dx.$$ ...
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### Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
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### The Bessel function and finding expression

The Bessel Function $J_v$ of the first kind of order $v$ can be defined by the series expresion $$J_v(x)=\sum_{n=0} ^{\infty} \frac{(-1)^n}{n!\Gamma{(1+v+n)}}\left(\frac{x}{2}\right)^{2n+v}$$ (i) if ...
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### What is the motivation behind the Bessel function of second kind

I am studying Bessel function and found the good reference by G.N. Watson At some point in page 58 he introduces the following expression due to Hankel: \begin{eqnarray} \lim_{\nu \to n} \frac{J_{\...
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### How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
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### Find a recurrence relationship for the following :

Find a recurrence relationhip for $a_{n}$: $a_{n}=\dfrac {2n+1}{2}\int^{1}_{-1}f\left( x\right) P_{n}\left( x\right) dx$ Where $f\left( x\right)= e^{-x}$ I have done it many times and keep ...
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### How is the following integral related to confluent hypergeometric functions?

I am solving an integral that appears in a physics paper. $$-\int_0^{\infty}dt\,\frac{e^{-t}}{t}\bigg[\bigg(1+\frac{3}{N}t\bigg)^N-1\bigg]$$ The paper does not give the full solution, it only gives ...
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### Particular values of the Riemann zeta function.

On the wikipedia, near the bottom of the "Specific Values" section, there is a statement that bothers me. $$\zeta(-13)=\zeta(-1)$$ Firstly, it is well noted that the summations must be evaluated ...
I try to show that $$\sum _{k=1}^{\infty } k^{36} \text{sech}(\pi k)=\frac{41222060339517702122347079671259045}{137438953472}+\frac{i \left(\psi _{e^{\pi }}^{(36)}\left(1-\frac{i}{2}\right)-\psi _{... 1answer 59 views ### Evaluate the indefinite integral \int \frac{t\sin at}{b^2+t^2}dt It is known DLMF (25.2.8) that for \Re s>0 and for integers N\geq 1$$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$where \... 3answers 232 views ### Fibonorial of a fractional or complex argument Let F(n) denote the n^{\text{th}} Fibonacci number^{[1]}$$\!^{[2]}$$\!^{[3]}. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument:$$F(z)=\left(\phi^...
In this blog entry, they give this ridiculous complicated expression for the first derivative of the Bessel function $J_n(x)$ that uses higher hypergeometric functions. I can't believe that a ...