Tagged Questions
4
votes
1answer
98 views
Integral using residue theorem (maybe)
I came across the following integral in a book (Kato's Perturbation Theory for Linear Operators, $\S$3.5):
$\int_{-\infty}^\infty (a^2+x^2)^{-n/2}\,dx$
where $n$ is a non-negative integer and $a$ is ...
1
vote
2answers
301 views
How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?
How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire?
Could you give me some scratch proof?
3
votes
0answers
100 views
Satisfying a Differential Equation and complex Laguerre
I have the following problem
Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
0
votes
0answers
96 views
Concept of Residue Cancellation
I am trying to understand how to apply the residue theorem to solve
$\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) ...
