# 梅林变换

（重定向自Mellin变换

${\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)dx.}$

${\displaystyle \left\{{\mathcal {M}}^{-1}\varphi \right\}(x)=f(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds.}$

## 與其他變換之關係

### 雙邊拉普拉斯變換

${\displaystyle \left\{{\mathcal {B}}f\right\}(s)=\left\{{\mathcal {M}}f(-\ln x)\right\}(s)}$

${\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\left\{{\mathcal {B}}f(e^{-x})\right\}(s)}$

### 傅立葉變換

${\displaystyle \left\{{\mathcal {F}}f\right\}(-s)=\left\{{\mathcal {B}}f\right\}(-is)=\left\{{\mathcal {M}}f(-\ln x)\right\}(-is)\ }$

${\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\left\{{\mathcal {B}}f(e^{-x})\right\}(s)=\left\{{\mathcal {F}}f(e^{-x})\right\}(-is)\ }$

## 範例

### Cahen–Mellin 積分

${\displaystyle e^{-y}={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }\Gamma (s)y^{-s}\;ds}$

### 數論

${\displaystyle \Re (s+a)<0}$

${\displaystyle f(x)={\begin{cases}0&x<1\\x^{a}&x>1\end{cases}}}$

${\displaystyle {\mathcal {M}}f(s)=-{\frac {1}{s+a}}}$

## 圓柱坐標系下的拉普拉斯算子

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)}$

${\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}}$

${\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}}$

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)=f_{rr}+{\frac {f_{r}}{r}}}$
${\displaystyle {\mathcal {M}}\left(r^{2}f_{rr}+rf_{r},r\to s\right)=s^{2}{\mathcal {M}}\left(f,r\to s\right)=s^{2}F}$

${\displaystyle r^{2}f_{rr}+rf_{r}+f_{\theta \theta }=0}$

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}=0}$

${\displaystyle F_{\theta \theta }+s^{2}F=0}$

${\displaystyle F(s,\theta )=C_{1}(s)\cos(s\theta )+C_{2}(s)\sin(s\theta )}$

${\displaystyle f(r,-\theta _{0})=a(r),\quad f(r,\theta _{0})=b(r)}$

${\displaystyle F(s,-\theta _{0})=A(s),\quad F(s,\theta _{0})=B(s)}$

${\displaystyle F(s,\theta )=A(s){\frac {\sin(s(\theta _{0}-\theta ))}{\sin(2\theta _{0}s)}}+B(s){\frac {\sin(s(\theta _{0}+\theta ))}{\sin(2\theta _{0}s)}}}$

${\displaystyle {\mathcal {M}}^{-1}\left({\frac {\sin(s\varphi )}{\sin(2\theta _{0}s)}};s\to r\right)={\frac {1}{2\theta _{0}}}{\frac {r^{m}\sin(m\varphi )}{1+2r^{m}\cos(m\varphi )+r^{2m}}}}$

${\displaystyle m={\frac {\pi }{2\theta _{0}}}}$

${\displaystyle f(r,\theta )={\frac {r^{m}\cos(m\theta )}{2\theta _{0}}}\int _{0}^{\infty }\left\{{\frac {a(x)}{x^{2m}+2r^{m}x^{m}\sin(m\theta )+r^{2m}}}+{\frac {b(x)}{x^{2m}-2r^{m}x^{m}\sin(m\theta )+r^{2m}}}\right\}x^{m-1}\,dx}$

## 参考文献

• Galambos, Janos; Simonelli, Italo. Products of random variables: applications to problems of physics and to arithmetical functions. Marcel Dekker, Inc. 2004. ISBN 0-8247-5402-6.