# 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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I have not frequented the board lately due to being very busy. But, I ran across what I think is an interesting problem. $\displaystyle \int_{0}^{k}(tx+1-x)^{n}dx$, where $n\in \mathbb{Z}^{+}$ and $... 1answer 3k views ### Steps in evaluating the integral of complementary error function? Could you please check the below and show me any errors? $$\int_ x^ \infty {\rm erfc} ~(t) ~dt ~=\int_ x^ \infty \left[\frac{2}{\sqrt\pi} \int_ t^ \infty e^{-u^2} du \right]\ dt$$ If I let dv=dt and ... 1answer 327 views ### Solving Shallow water Equations with Hermite polynomials I have problem with solving the shallow water equations near beaches to achieve the wave run-up over the shore line. The main equation is $$\frac{d^2\eta}{dt^2} + \frac{d}{dx}\left(h\frac{d\eta}{dx}\... 3answers 447 views ### Theory of the Mathieu Operator How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ... 1answer 249 views ### Zeta function identity How does one prove the zeta function identity$$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$1answer 235 views ### Heuristic\iterated construction of the Weierstrass nowhere differentiable function. I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example ... 1answer 161 views ### What is the reason to use hypergeometric functions? I would be grateful if anyone could explain the purpose of using hypergeometric functions. If a function exists in closed form, e.g. \sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert ... 1answer 305 views ### Maclaurin series involving an elliptic integral I have been asked to find the Maclaurin series expansion of a term involving an elliptic integral, I would be grateful for any help as I am unsure as to how to even start this question. The term I ... 1answer 863 views ### Recursive solutions to linear ODE. When finding the solutions to the simple ODE$$ y'- mxy= x^n \text{ ; } y(0) = 0$$I found the following: Let P_n be the particular solution for each integer exponent n. Then if we define$$... 1answer 305 views ### Integration and$\gamma(n,x)$It isn't hard to prove that: $$\int_0^x e^{-t} {t^n} dt = n! \cdot e^{-x}\left( e^x-\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$ Or put in a different way: $$\int_0^x e^{-t} \frac{t^n}{n!} dt = e^{-x}\... 1answer 353 views ### Integrals \int\limits_0^t {{e^u}\log u\operatorname d \!u} and \int\limits_0^t {{e^{ - u}}\log u\operatorname d \!u} Ok, I want to find$$\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$$... 2answers 1k views ### Spherical Bessel Zeros I was wondering if there is a known closed form solution for the zeros of the spherical Bessel functions. While doing a quantum assignment, I came across them as a solution for the spherical infinite ... 3answers 822 views ### Weierstrass \wp function doubly periodic I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ... 2answers 1k views ### The roots of Hermite polynomials are all real? The Hermite polynomials are defined as$$H_n(x)=(-1)^n e^{x^2}\dfrac{d^n}{dx^n}e^{-x^2}.$$How does one prove that all the roots of the Hermite polynomial are real? 1answer 704 views ### Integral representation of modified bessel function How can I obtain$$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 .$$... 1answer 391 views ### Verifying identities for Riemann zeta function I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ... 1answer 801 views ### Higher Order Trigonometric Function Once in a time, I had to work with functions that have the following Taylor series expansion:$$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$... 1answer 2k views ### Associated Legendre polynomials of fractional order My question concerns the associated or generalized Legendre polynomials. They are labeled by two numbers m and l; i.e., P_l^m(x), for x \in [-1,1]. Usually one assumes that m and l are ... 1answer 156 views ### distribution function Let(X,\mu) be a measure space. f:X \to \mathbb{R} be a measure function. For every t\in \mathbb{R} the distribution function F of f is defined as F(t)=\mu\{x \in X:f(x)<t\}. I have ... 1answer 247 views ### Create 'smooth breakpoint function' by using integral? Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a ... 3answers 105 views ### Oddities in the Definition of IntegralCosinus {\rm Ci}(x) Reading the defintion of the IntegralCosinus$$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$I wonder what happens, if I to split the function in the integral:$$ \begin{... 2answers 572 views ### Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ... 1answer 158 views ### Solving an integral using generating functions, coefficients of equivalent series don't match!? Please don't be frightened by the length of this, I just wanted to provide ample detail. If you want, you can skip the derivation and go straight to the result at the bottom. I have $$\begin{... 3answers 935 views ### Limit of Zeta function I'm looking for a reference for (or an elementary proof of)$$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$Thanks for your help. 1answer 952 views ### On the completeness of the generalized Laguerre polynomials I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space L^2(\mathbb{R}). 1. Orthonormality \int_0^{\infty} e^{-x}x^kL_n^k(x)L_{m}^k(x)dx=\... 1answer 430 views ### Advanced application of the Binomial Theorem I'm trying to solve the following integral:$$ \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 $$Where C_{n}^{\lambda}(x) is a ... 1answer 200 views ### real-value solution I have this integral$$\int\frac{dz}{\sqrt{(z^{2}-\rho^{2})(\lambda^{2} - z^{2})}}$$and parameters obey the following conditions$$z= \exp[k\varphi],\lambda^{2} = \frac{b + \sqrt{b^{2} - 4ac}}{... 1answer 116 views ### Problem with the Limits of the Hypergeometric Series I'm new to the hypergeometric series, and I'm trying to decipher a proof in which the author identifies a particular finite sum as a a hypergeometric series. The particular summation is: $$\begin{... 1answer 162 views ### how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian) I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma \left(\... 1answer 262 views ### Closed form formula for series involving derivatives of reciprocal gamma function How to get closed form for the sum \displaystyle{\sum\limits_{k = 1}^\infty {\frac{{{p^k}}} {{\left( {2k} \right)!!}}\frac{{{d^k}}} {{d{s^k}}}{{\left. {\frac{1} {{\Gamma \left( s \right)}}} \... 1answer 1k views ### How do you prove Gautschi's inequality for the gamma function? A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi:$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ... 1answer 315 views ### Orthogonality of the Gegenbauer Polynomials Typically the orthogonality relation for the Gegenbauer polynomials is given as: $$\int_{-1}^{1}C_{n}^{\alpha}(x)C_{m}^{\alpha}(x)\cdot(1-x^2)^{\alpha-1/2}dx=\frac{\pi2^{1-2\alpha}\Gamma(2\alpha+n)}... 3answers 446 views ### Reference requests: Jitsuro Nagura I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev \psi function says that ... 1answer 82 views ### Trigonometric term in digamma function \psi_{0}(-n) Solutions to expressions s.a.$$ S(n)=\sum_{k=1}^{n}\frac{1}{k-r} = \psi_{0}(n-r+1)- \psi_{0}(1-r), $$involves digamma function. For positive values it has the largest term O(\log(n)), but ... 1answer 150 views ### Integration about standard normal Let N(x) denote the cdf of standard normal and n(x) denote the pdf of standard normal. How to evaluate the integral \int\limits_{-\infty}^\infty N(a+x) n(x) \mathrm{d} x ? Thanks a lot! 1answer 553 views ### Upper bound on differences of consecutive zeta zeros The average gap \delta_n=|\gamma_{n+1}-\gamma_n| between consecutive zeros (\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i) of Riemann's zeta function is \frac{2\pi}{\log\gamma_n}. There are many ... 1answer 119 views ### Looking for a magnifyer function I am looking for a function f having the following characteristics: f defined on [0,1] f(0)=0 f(1)=1 \forall x \in ]0,1[, x <f(x) < 1 f differentiable on ]0,1] f'>0 ... 1answer 321 views ### Proof of an inequality under Equivalence of Weierstrass and Euler's Definitions of Gamma Function I am using my lecturer's notes on Special Functions. When dealing with Gamma Functions under the title Equivalence of Wierstrass and Euler's Definitions, he has used a lemma (with no proof). I tried ... 1answer 819 views ### Inverse function of \operatorname{li}(x) over x>\mu? How can I get the inverse function of \operatorname{li}(x) over x>\mu? Where$$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$is the so-called logarithmic integral, and$\operatorname{li}...
The (real) digamma function is new to me and I notice that under some circumstances a sum of $\log\Gamma$ functions will have a derivative consisting of a sum of digamma functions that converge to a ...
For suitable choice of a one-parameter family of functions $\{ g_w:\mathbb{R} \rightarrow \mathbb{C} \}_{w\in \mathbb{R}}$, the following two statements are equivalent (modulo sets of measure $0$): \$\...