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

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21
votes
1answer
567 views

Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — ...
1
vote
0answers
149 views

Orthogonal polynomial interpolation of a function

I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
3
votes
0answers
207 views

How did Bessel functions come to be denoted by $J_n$?

The $n$th Bessel function of the first kind is usually denoted $J_n(x)$. Where did the use of the letter $J$ to indicate the Bessel function come from?
1
vote
2answers
425 views

Integral of the fractional part of $\frac1x$ multiplied by $x$ on interval $(a,b), a\ge 0$.

I'm interested in finding the value of the integral of $\left\{\frac{1}{x}\right\}\cdot x$ (the fractional part of $\dfrac{1}{x}$ multiplied by $x$) on the interval $(a,b), a\ge 0$ the integral of ...
11
votes
1answer
576 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
3
votes
3answers
928 views

Continuous function with local maxima everywhere but no global maxima

Can there be such a function: $f \colon \mathbb R \to \mathbb R$ is continuous and non-constant. It has a local maxima everywhere, i.e., for all $x \in \mathbb R$ there is some $\delta_x>0$ such ...
6
votes
1answer
316 views

What's the sum of this power series?

What's the sum of this power series? $$f_k(x)=1-\frac{x^2}{k}+\frac{x^4}{k(k+1)\cdot2!}-\frac{x^6}{k(k+1)(k+2)\cdot3!}+\ldots$$ I'm just helping someone, I'm not good at math! :\
4
votes
2answers
316 views

Determination of inverse laplace transform using primitive functions

In How can you prove that a function has no closed form integral?, the accepted answer points to http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf where one can find a corollary by Liouville ...
14
votes
5answers
951 views

Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$

Can you help prove the functional equation: $$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$ Specifically, I am looking for a solution using complex ...
6
votes
2answers
1k views

Integral involving Modified Bessel Function of the First Kind

Why is this true? $$ \int_0^\infty e^{-\frac{1}{2}(b^2+x^2)} I_0(bx) x \,dx = 1 $$ Note that $I_0(x)$ is a modified bessel function of the first kind. The difficulty for me lies in a) translating ...
7
votes
1answer
304 views

Is there a gamma-like function for the q-factorial?

I'm looking at quantum calculus and just trying to understand what is going with this subject. Looking at the q-factorial made me wonder if this function could take all real or even complex numbers in ...
0
votes
2answers
428 views

Proving a Laguerre polynomial integral

After a fair bit of effort, I managed to prove that $$\int_0^\infty t^\alpha \exp(-t) L_n^{\alpha+1}(t)\mathrm dt=\Gamma(\alpha+1)$$ where $L_n^\alpha (t)$ is a generalized Laguerre polynomial, with ...
6
votes
1answer
857 views

How to decompose displaced Hermite-Gauss function into higher order HGs?

The Hermite-Gauss functions appear commonly in physics. These functions are formed from the product of a Hermite polynomial and a Gaussian: $$ u_n(x) = \left(\frac{2}{\pi w_0^2}\right)^{1/4} ...
3
votes
2answers
394 views

expression for the sum involving digamma function

I got this answer from WolframAlpha. Does anyone know how even to approach it to obtain the solution using digamma function. Please don't solve it, just show me in the right direction! $$ ...
4
votes
1answer
201 views

Closed form for some integrals related to the complementary error function

While studying the use of the trapezoidal rule for numerically evaluating the complementary error function $\mathrm{erfc}(z)$, the following integrals showed up when I was trying to derive expressions ...
3
votes
3answers
1k views

How to solve $n$ for $n^n = 2^c$?

How to solve $n$ for $n^n = 2^c$? What's the numerical method? I don't get this. For $c=1000$, $n$ should be approximately $140$, right?
10
votes
3answers
641 views

How does Lambert's W behave near ∞?

How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of $$\frac{W(x)}{\ln(x)}$$ near $\infty$ (but along the positive real line, if that ...
10
votes
2answers
524 views

How to show $\sum_{n=-\infty}^\infty J_n J_{n+m} = \delta(m)$?

The following is an identity concerning the Bessel functions of the first kind $J_n(x)$ for integers $n$ and $m$: $$\sum_{n=-\infty}^\infty J_n(x) J_{n+m}(x) = \delta(m)$$ where $\delta(x)$ is the ...
3
votes
1answer
637 views

Approximation for Lambert W function near zero

I am looking for a good approximation for the $W_0$ branch of the Lambert $W$ function. I am looking for values $0 < x < e$ only, so I expect something simpler than the general Taylor expansion. ...
8
votes
1answer
454 views

The uniqueness of the Gamma Function

It is a theorem that any function $f$ defined for positive real numbers satisfying $f(1)=1$ $f(x+1)=x\cdot f(x)$ $f$ is log convex is identically equal to the gamma function. (Condition 2 means ...
8
votes
1answer
489 views

Monotonic behavior of a function

I have the following problem related to a statistics question: Prove that the function defined for $x\ge 1, y\ge 1$, ...
7
votes
1answer
374 views

On the Bessel function $J_n(z)$ for high $z$, with respect to $n$

Plotting the Bessel functions of the first kind $J_n(z)$ versus $n$ for some fixed $z\gg1$, it appears that there is a sharp cutoff just before $n=z$. Three questions: What is a reference ...
12
votes
1answer
385 views

Behaviour of the series $\exp_p(x)=\sum_{k=0}^{\infty}\frac{x^k}{(k!)^p}$ depending on $p\approx 2$?

Note:This is more a math-recreational question Consider the series $\exp_p(x)=\sum_{k=0}^{\infty}\frac{x^k}{(k!)^p}$ which is some systematic modification of the exponential function. It's ...
3
votes
1answer
107 views

Validity of a q-series theorem

Define the $q$-analog $(a;q)_n = \prod_{k=0}^n \left(1 - aq^k\right)$. I want to prove the identity $\frac{(q^2;q^2)_\infty}{(q;q)_\infty}=\frac{1}{(q;q^2)_\infty}$. I viewed the LHS this way: ...
0
votes
2answers
905 views

Find intersection of linear and logarithmic lines

I have equations for two lines, one of which is linear and the other is logarithmic, ie: $$y = m_1 x + c_1$$ $$y = m_2 \cdot \ln(x) + c_2$$ ..and I need to find out where (if at all) these lines ...
6
votes
1answer
285 views

Elliptic functions and Weierstrass $\wp$-function

Question that seems pretty easy, but I can't formalize it: Let $L \subset C$ be a lattice, and $f(z)$ be an elliptic function for $L$, that is a meromorphic function so that $f(z+w) = f(z)$ for all ...
10
votes
3answers
442 views

Erroneous numerical approximations of $\zeta\left(\frac{1}{2}\right)$?

By definition of the Riemann Zeta Function, $$\zeta\left(\frac{1}{2}\right) = \sum_{n=1}^\infty \frac{1}{\sqrt{n}}.$$ Since $\forall n \geq 1 : \frac{1}{\sqrt{n}} \geq \frac{1}{n}$, we have that for ...
3
votes
0answers
183 views

Solid angle spanned by disc/rewriting expression with elliptic integrals

The solid angle spanned by a disc of unit radius, as observed from a point $(r,z)$ at a distance $z>0$ above a point in the disc plane with at distance $r>0$ to the center, can be expressed as ...
5
votes
1answer
358 views

Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?

In my recent explorations I stumbled upon the following series $$ \vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...
2
votes
0answers
47 views

Deriving functions for empiracal distributions -very applied mathatics

First I am a new user of this site. Second my math background is very limited, although I do have a lot of experience in applied statistics. Component or piece part failures on high value parts($1000 ...
5
votes
1answer
512 views

what does the “L” in “L-function” stand for?

I haven't been able to find a reference that tells what word (if a word) the L is short for.
6
votes
2answers
331 views

What is Eulerian?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld: $$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$ It is interesting because it seems that roots of any ...
7
votes
6answers
5k views

Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
6
votes
1answer
1k views

Is there an Inverse Gamma $\Gamma^{-1} (z) $ function?

Since $\Gamma$ is not one to one over the complex domain, Is it possible to define some principal values ( analogues to Principal Roots for the Root function ) so we can have a $\Gamma^{-1} (z)$ ...
19
votes
3answers
721 views

Why isn't the gamma function defined so that $\Gamma(n) = n! $?

As a physics student, I have occasionally run across the gamma function $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$ when we want to generalize the concept of a factorial. ...
5
votes
3answers
1k views

Integral Representations of Hermite Polynomial?

One of my former students asked me how to go from one presentation of the Hermite Polynomial to another. And I'm embarassed to say, I've been trying and failing miserably. (I'm guessing this is a ...
2
votes
0answers
349 views

How can I integrate a Bessel function divided by a “shifted” value?

Sorry to ask yet another "how do I do this integral" question! But I've really been having a hard time with this one. $$\int_0^\infty \frac{J_0(x)}{|C^2 - x^2|}\mathrm{d}x$$ I've been through a lot ...
8
votes
3answers
7k views

How do you integrate a Bessel function? I don't want to memorize answers or use a computer, is this possible?

I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various ...
3
votes
2answers
356 views

Periodicity of EllipticPi

We are trying to implement transformations to evaluate the incomplete integral of the third kind $\Pi(n;\phi|m)$ for arbitrary inputs, and I can't find any references for how to calculate this ...
26
votes
4answers
2k views

When is an elliptic integral expressible in terms of elementary functions?

After seeing this recent question asking how to calculate the following integral $$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$ and some of the comments that suggested that it was an ...
2
votes
1answer
484 views

Is Riemann Zeta Function symmetrical about the real axis?

From wikipedia, http://en.wikipedia.org/wiki/Riemann_zeta_function "Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
8
votes
1answer
622 views

Stirling-type formula for the logarithmic derivative of the Gamma function

How may one go about proving $\displaystyle\frac{\Gamma'(s)}{\Gamma(s)}=O(\log|s|)$, (away from the poles) directly? By a direct proof, I mean not to go through the usual Stirling formula with ...
9
votes
5answers
736 views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
4
votes
1answer
452 views

How can I prove this identity involving the digamma function?

I'm trying to prove an identity involving the digamma function $\psi(z)$, but I can't seem to figure out a way to do it. Can anyone help me out? The identity is $$\psi\left(\frac{m}{2} + iy\right) + ...
4
votes
2answers
777 views

Domain of the Gamma function

I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral: $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$ converges. I started by ...
14
votes
3answers
2k views

What is a special function?

When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged ...
3
votes
2answers
400 views

Is it problem of Mathematica or my own?

The following is a plot comparing Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma[x]: In theory the blue and the red ...
6
votes
1answer
1k views

Some questions about the gamma function

Show that $\Gamma(y) = \int_0^{\infty}{e^{-x}x^{y-1}\,dx}$ is finite for $y>0$ both as an improper Riemann integral and as a Lebesgue integral. Show $\Gamma'(y) = ...
10
votes
1answer
304 views

density of roots of a family of polynomials: $(1-x^2)^{v+n}$

My research has brought me to the following, very general problem. Given a fixed, but arbitrary, natural number, $\displaystyle v$, consider the following family of polynomials: The $\displaystyle ...
16
votes
5answers
4k views

Roots of Legendre Polynomial

I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials. Are the roots always simple (i.e., multiplicity $1$)? ...