- #1
rbwang1225
- 118
- 0
\documentclass[12pt,a4paper,twoside]{article}
\usepackage{amssymb,amsmath,amsbsy,amscd,fancyhdr}
\usepackage[mathscr]{eucal}
\usepackage{tikz}
\usepackage{graphicx,cite,xcolor,comment,soul,ulem}
\begin{document}
I met some problem in the following integration; does anyone know how to solve it?
\begin{equation}
\frac{2m}{h^2}\frac{1}{4\pi}\int{d^3xe^{i\mathbf{k\cdot r}}\frac{V_0e^{-\mu r}}\mu r}e^{ikx}.\
\end{equation}
The key part that can not be solved is:
\begin{equation}
\int_0^{\infty}\int_0^{\pi}e^{(ik-\mu) r}e^{ikr\cos\theta}\sin\theta d\theta dr.\
\end{equation}
I found it can be treated as a Laplace transform of
\begin{equation}
\frac{e^{ikr}\sin(kr)}r
\end{equation}
And the solution is
\begin{equation}
i\tanh^{-1}[\frac{k}{k+i\mu}].
\end{equation}
Any help would be appreciated!
\end{document}
\usepackage{amssymb,amsmath,amsbsy,amscd,fancyhdr}
\usepackage[mathscr]{eucal}
\usepackage{tikz}
\usepackage{graphicx,cite,xcolor,comment,soul,ulem}
\begin{document}
I met some problem in the following integration; does anyone know how to solve it?
\begin{equation}
\frac{2m}{h^2}\frac{1}{4\pi}\int{d^3xe^{i\mathbf{k\cdot r}}\frac{V_0e^{-\mu r}}\mu r}e^{ikx}.\
\end{equation}
The key part that can not be solved is:
\begin{equation}
\int_0^{\infty}\int_0^{\pi}e^{(ik-\mu) r}e^{ikr\cos\theta}\sin\theta d\theta dr.\
\end{equation}
I found it can be treated as a Laplace transform of
\begin{equation}
\frac{e^{ikr}\sin(kr)}r
\end{equation}
And the solution is
\begin{equation}
i\tanh^{-1}[\frac{k}{k+i\mu}].
\end{equation}
Any help would be appreciated!
\end{document}