Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

Higher transcendental functions are frequently termed special functions. These functions were studied extensively in the eighteenth and nineteenth centuries-by Gauss, Euler, Abel, Jacobi, Weierstrass, Riemann, Hermite, Poincare, and other leading mathematicians of the day. Although many of the functions that they treated were quite recondite and are no longer of much interest today, others (such as the Riemann zeta function, the gamma function, and elliptic functions) are still intensively studied.

Before asking, please make sure that you define your notation very precisely, as
1. not everybody is familiar with the notation for special functions; and
2. a lot of special functions have different notational conventions, depending on the paper/book.

You might want to first check if the special function you are considering is discussed in Abramowitz and Stegun, the Digital Library of Mathematical Functions, or the Wolfram Functions site.

References:

https://en.wikipedia.org/wiki/Special_functions

https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms

"Special Functions and Their Applications" by R. Silverman

4626 questions
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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 for the truncation error: $$\int_0^\pi…
Juan Joder
  • 139
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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) + \psi\left(\frac{m}{2} - iy\right) =…
David Z
  • 3,539
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Expressing upper incomplete gamma function of half-integer order in terms of gamma function?

N. M. Temme, "Special Functions" (Wiley 1996) gives the following expression that expresses the upper incomplete gamma function in terms of the ordinary gamma function, for integer orders: $$ \Gamma(n,z) = \Gamma(n) e^{-z} \sum_{m=0}^{n-1}…
njuffa
  • 1,768
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A singularity of hypergeometric functions

Do generalized hypergeometric functions $${}_p F_q(a_1,\ldots,a_p; b_1, \ldots,b_q; z) $$ with $p = q+1$ always possess a singularity at $z=1$, independent of the their parameters $a_1,\ldots,a_p$ and $b_1,\ldots,b_q$ under the provision that all…
clickmock
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2 answers

About Rayleigh's formula

How to use $\displaystyle j_n(x)=(-1)^nx^n\left(\frac{1} {x} \frac{d} {dx}\right)^n \left(\frac{\sin x}{x}\right)$? for example, to find $j_3(x)$
tweelly
  • 117
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Verify the expansion

$\displaystyle \int_{0}^{x}J_\nu(t)dt=2\sum_{n=0}^{\infty}J_{\nu +2n+1}(x)$ Hint: Using the following recurrence relations show that both sides have the same derivative. $\displaystyle J_{\nu-1}(x)+J_{\nu+1}(x)=\frac…
Bon Les
  • 75
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Normalization of the Bessel function

I would greatly appreciate assistance with the following problem. show: $$\int _0 ^\infty J_n(x)dx = 1; \forall n \in \mathbb{N}^+$$ for $J_o,$ use $$\mathscr{L}{J_o(at)} = \int _0 ^\infty e^{-pt}J_o(at)dt = (p^2 + a^2)^{- \frac{1}{2}}$$ By setting…
Cactus BAMF
  • 1,047
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How to prove Gegenbauer's addition theorem?

How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\over 2 r_1 r_2}\int_{|r_1 - r_2|}^{r_1+r_2} e^{-{r\over D}} P_k\left(r_1^2 - r^2 + r_2^2 \over 2 r_1 r_2 \right) d r= $$ $$ =(2k+1) …
Ondřej Čertík
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Does there exist a nicer form for $\beta(x + a, y + b) / \beta(a, b)$?

I have the expression $$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$$ where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$. I have a feeling this should have a closed-form which is intuitive and makes less…
singelton
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2 answers

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 function with phase $\phi$ less than zero or greater…
Will Robertson
  • 175
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1 answer

Value of a scaled Bessel function for negative argument

Is the function $\hat{i}_0(x) = e^{-|x|} \sqrt{\frac{\pi}{2x}} I_{\frac{1}{2}}(x)$ positive or negative for negative $x$? $I_{\alpha}(x)$ above is a modified Bessel function. Here are my arguments. Considering that $I_{\frac{1}{2}}(x) =…
vitaut
  • 143
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Is this series expressible in terms of Gauss' hypergeometric function?

How we can express this series $$F(z)=\sum_{n=0}^\infty \frac{z^n}{(a)_nn!}$$ in terms of Gauss' hypergeometric function? where $(a)_n$ denotes the Pochhammer symbol. Thanks in advance
MAK
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Prove or disprove that Weierstrass $\wp$ function is holonomic

Recall that a holonomic function $f$ (say over $\mathbb C$) is one that is a solution to a differential equation of the form: $$p_0(z)f(z)+p_1(z)f'(z)+p_2(z)f''(z)+\dots+p_k(z)f^{(k)}(z) = 0$$ where the $p_i$ are polynomials in $\mathbb C[z]$ for…
quantum
  • 1,645
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What special function is this?

Assume that $\zeta$ is a positive real number and $a = \frac{2 \pi}{\alpha_{\text{max}}}$ for $0 < \alpha_{\text{max}} < \frac{\pi}{2}$. In other words $a > 4$. Is there a special function that when evaluated in a certain point is equal…
JT_NL
  • 14,514
3
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1 answer

Is anything known about $2\pi$ integer multiple arguments of the cosine integral?

I'm interested in $\text{Ci}(2\pi n)$ for integers $n\geq 1$. As the graph below shows, as $n$ increases the cosine integral seems to (strictly?) monotonically decrease. I've looked online but can't find much, and I'm wondering - is there a closed…
pshmath0
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