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 does the Gamma function interpolate $(n-1)!$?

Why does the Gamma function interpolate $(n-1)!$ and not $n!$ instead? What is the historical reason?
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Reasoning about the gamma function using the digamma function

I am working on evaluating the following equation: $\log\Gamma(\frac{1}{2}x) - \log\Gamma(\frac{1}{3}x)$ If I'm understanding correctly, the above is an increasing function which can be demonstrated ...
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Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
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Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
725 views

Why $p$-adically interpolate?

I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
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Prove that $\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}$

Does anybody know how to prove this identity? $$\int_0^{\pi/2} \cos^{p+q-2}(\theta) \cos((p-q)\theta)d\theta = \frac{\pi}{(p+q-1)2^{p+q-1}B(p,q)}\quad p+q>1,q<1$$ $B(x,y)$ denotes Beta ...
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Verification of integral over $\exp(\cos x + \sin x)$

I found the following integral in a paper I was reading: \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \exp\left(a \cos x + b \sin x\right) dx = I_0\left(\sqrt{a^2+b^2}\right), \end{...
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How to solve $x \geq \frac{y}{z-\ln{x}}$ for positive variables?

How can you solve $x \geq \frac{y}{z-\ln{x}}$ for $x$ when the variables are real positive values? I am only really interested in the case where the values are large and $z > \ln x$. How ...
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Bessel function confirmation

I'm trying to see that $J_0(x)$ is indeed a solution for the Bessel equation $x^2y''+xy'+x^2y=0$, so: $$J_0(x)=\sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(k!)^22^{2k}}$$ Pluging it in the equation and and ...
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The double integral $\int_0^1 \int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s \, dx \, dy$

$$\int_0^1 \int_0^1 \frac{1-x}{1-xy}(-\log(xy))^s \, dx \, dy=\Gamma(s+2)\left(\zeta(s+2) -\frac{1}{s+1}\right) \quad \Re(s)>-2$$ How to prove this identity?
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Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
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Roots of the equation $I_1(b x) - x I_0(b x) = 0$

I'm interested in the roots of the equation: $I_1(bx) - x I_0(bx) = 0$ Where $I_n(x)$ is the modified Bessel function of the first kind and $b$ is real positive constant. More specifically, I'm ...
Numerical approximation of the modified Bessel function $I_0$ with radical argument for integration purposes
I have to numerically calculate the following definite integral $$\int_{\alpha}^{\beta}I_0(a\sqrt{1-x^2})dx$$ for different values of $\alpha$ and $\beta$, where $a$ has a value of, say, $30$. I'm ...