# 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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### Integrals $\int\limits_0^t {{e^u}\log u\operatorname d \!u}$ and $\int\limits_0^t {{e^{ - u}}\log u\operatorname d \!u}$

Ok, I want to find $$\int\limits_0^t {{e^u}\log udu}$$ and $$\int\limits_0^t {{e^{ - u}}\log udu}$$ I'm thinking as follows $$d\left( {{e^u}\log u} \right) = {e^u}\log udu + \frac{{{e^u}}}{u}du$$...
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### Spherical Bessel Zeros

I was wondering if there is a known closed form solution for the zeros of the spherical Bessel functions. While doing a quantum assignment, I came across them as a solution for the spherical infinite ...
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### Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
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### The roots of Hermite polynomials are all real?

The Hermite polynomials are defined as $$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$ How does one prove that all the roots of the Hermite polynomial are real?
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### Integral representation of modified bessel function

How can I obtain $$I_n(x)= \frac{1}{2\pi}\int_{0}^{2\pi}e^{in\theta}e^{x \cos\theta}\,d\theta ,$$ from the integral $$J_n (x) =\frac{1}{2\pi} \int_{0}^{2\pi}e^{-in\theta}e^{ix \sin\theta}\,d\theta .$$ ...
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### Verifying identities for Riemann zeta function

I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
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### Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}.$$ ...
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### Associated Legendre polynomials of fractional order

My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers $m$ and $l$; i.e., $P_l^m(x)$, for $x \in [-1,1]$. Usually one assumes that $m$ and $l$ are ...
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### distribution function

Let$(X,\mu)$ be a measure space. $f:X \to \mathbb{R}$ be a measure function. For every $t\in \mathbb{R}$ the distribution function $F$ of $f$ is defined as $F(t)=\mu\{x \in X:f(x)<t\}.$ I have ...
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### Create 'smooth breakpoint function' by using integral?

Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a ...
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### Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$\lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.
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### On the completeness of the generalized Laguerre polynomials

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \int_0^{\infty} e^{-x}x^kL_n^k(x)L_{m}^k(x)dx=\...
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### Advanced application of the Binomial Theorem

I'm trying to solve the following integral: $$\int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx$$ Where $C_{n}^{\lambda}(x)$ is a ...
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### how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma \left(\...
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### Proof of an inequality under Equivalence of Weierstrass and Euler's Definitions of Gamma Function

I am using my lecturer's notes on Special Functions. When dealing with Gamma Functions under the title Equivalence of Wierstrass and Euler's Definitions, he has used a lemma (with no proof). I tried ...