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The function f[z_]:=2^(i-k/2-v/2) (k+v)! Gamma[1+i,z] Pochhammer[-v, i]/(i! k! Pochhammer[-k-v,i]) for $i,v,k \in \mathbb{N}_0$ and $z\in \mathbb{R}^-$ is for some combinations of $i,v,k$ only define …

MMA does not calculate the limit, however the limit exists as seen in the approximative plot. Limit[HypergeometricPFQ[{-1/2}, {1/2, m/2}, x]/Gamma[m], m -> 0] How to achieve the limit? MMA 13.0

Could MMA solve analytically this integral by using a Dirac delta function or in another way? $$ f(x)=\int_{-\infty}^{\infty}\left(\frac{1}{3 t^2+1} e^\frac{-t^2+i t}{3 t^2+1}\right)e^{itx}{\rm d}t\ta …

How can I express in MMA a derivative if the variable is a function? Example: The first derivative of $x^4$ with respect to $x^2$, would be $\dfrac{\mathrm{d}{x^4}}{\mathrm{d}{x^2}}=2x^2$ ? D[x^4,x^2] …

MMA delivers: $$\int_0^a\sqrt{\frac{1}{x}-\frac{1}{a}}\cosh\left(\sqrt{x}\right)\textrm{d}x=\pi I_1(\sqrt{a})$$ where $a>0$ and $I_1$ is the modified Bessel function of $1^\textrm{st}$ kind and order …

MMA differently handles division by zero. How to control this behavior? In this case it delivers the expected error: i/(i + k) /. {i -> 0, k -> 0} (* Indeterminate *) but if you calculate the same usi …

The Harmonic number $H_{n-1}$ can be expressed by the Digamma function $\psi(n)$ together with the Euler-Mascheroni constant $\gamma\approx 0.5772$. $\psi(n)=H_{n-1}-\gamma$ In MMA $\psi(n)$ and $\g …

Why MMA delivers this result and how to interpret or reformulate it? Limit[k Sum[1/i, {i, 1, k - 1}], k -> 0] I would expect 0 as it is an empty sum. MMA 12.1 Edit: Both expressions below deliver t …

For definite integrals MMA gives identities in terms of Meijer G-functions, e.g. $\begin{align}\sqrt{\pi}\int_0^\infty \textrm{e}^{-4x/t^2-t}\ \textrm{d}t &= G_{0,\,3}^{3,\,0} \left( x\left. \right …

Thanks to user Mariusz Iwaniuk the answer can be found easily: Series[Gamma[1/2 + x/2]/Gamma[x/2], {x, Infinity, 0}] returns $$\sqrt{\frac{x}{2}}-\frac{1}{4\sqrt{2x}}+\mathcal{O}\left(\frac{1}{x}\righ …

Can MMA find limits if the limit can be expressed as a function? Example: $$\lim_{x \to \infty}\frac{\Gamma\left(\frac{x+1}{2}\right)}{\Gamma\left(\frac{x}{2}\right)} =\sqrt\frac{x}{2}=\infty$$ $\\\\$ …

How could the following integration of a 6-dimensional Gaussian be achieved? Are there some techniques? (MMA code below) $$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty …