梅林变换

之所以伽马函数与积分变换的理论联系密切，是因为伽马函数同时是指数函数的拉普拉斯变换和幂函数的梅林变换，这也展示了两种积分变换之间的联系。

在任何維度的圓柱坐標系中，拉普拉斯算子總是會包含下式

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

例如，拉普拉斯算子在二維空間的極坐標表示法

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

或是在三維空間的柱坐標表示法

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

而利用梅林變換可以很簡單的處理此項

${\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)=f_{rr}+{\frac {f_{r}}{r}}}$

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

舉例來說，二維拉普拉斯方程的極坐標表示法具有以下形式

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

或是

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

利用梅林變換，可以轉換成一個簡諧振子的形式

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

通解為

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

若給定邊界條件

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

其梅林變換為

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

則通解可以寫成

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

最後利用逆變換以及卷積定理

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

其中

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

可以得到

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

因为具有尺度不变性，梅林变换广泛应用于计算机科学的算法分析。[1]对于纯虚输入，原函数函数的梅林变换与对其进行尺度伸缩后函数的梅林变换幅度相同。尺度不变性类似于傅里叶变换的时移不变性，即原函数与对其进行时移的函数的傅里叶变换幅度相同。这一性质对图像识别非常有用：当物体与摄像机的距离发生变化时，图像尺度会发生变化。

在量子力学（特别是量子场论）中，由于动量和位置之间存在傅里叶变化的关系，傅里叶空间被广泛应用。2011年，A. Liam Fitzpatrick、Jared Kaplan、João Penedones、Suvrat Raju和Balt C. van Rees证明了梅林空间在AdS/CFT对偶中具有类似的效用。[2][3][4]

下表展示了部分函数的梅林变化结果，在Bracewell (2000)与Erdélyi (1954)中可以找到。

梅林变化表

函数 ${\displaystyle f(x)}$	梅林变换 ${\displaystyle {\tilde {f}}(s)={\mathcal {M}}\{f\}(s)}$	收敛域	注释
${\displaystyle e^{-x}}$	${\displaystyle \Gamma (s)}$	${\displaystyle 0<\Re s<\infty }$
${\displaystyle e^{-x}-1}$	${\displaystyle \Gamma (s)}$	${\displaystyle -1<\Re s<0}$
${\displaystyle e^{-x}-1+x}$	${\displaystyle \Gamma (s)}$	${\displaystyle -2<\Re s<-1}$	一般来说， ${\displaystyle \Gamma (s)}$是${\displaystyle e^{-x}-\sum _{n=0}^{N-1}{\frac {(-1)^{n}}{n!}}x^{n},{\text{ for }}-N<\Re s<-N+1}$的梅林变换。[5]
${\displaystyle e^{-x^{2}}}$	${\displaystyle {\tfrac {1}{2}}\Gamma ({\tfrac {1}{2}}s)}$	${\displaystyle 0<\Re s<\infty }$
${\displaystyle \mathrm {erfc} (x)}$	${\displaystyle {\frac {\Gamma ({\tfrac {1}{2}}(1+s))}{{\sqrt {\pi }}\;s}}}$	${\displaystyle 0<\Re s<\infty }$
${\displaystyle e^{-(\ln x)^{2}}}$	${\displaystyle {\sqrt {\pi }}\,e^{{\tfrac {1}{4}}s^{2}}}$	${\displaystyle -\infty <\Re s<\infty }$
${\displaystyle \delta (x-a)}$	${\displaystyle a^{s-1}}$	${\displaystyle -\infty <\Re s<\infty }$	${\displaystyle a>0,\;\delta (x)}$ 是狄拉克函数。
${\displaystyle u(1-x)=\left\{{\begin{aligned}&1&&\;{\text{if}}\;0<x<1&\\&0&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}$	${\displaystyle {\frac {1}{s}}}$	${\displaystyle 0<\Re s<\infty }$	${\displaystyle u(x)}$ 是单位阶跃函数。
${\displaystyle -u(x-1)=\left\{{\begin{aligned}&0&&\;{\text{if}}\;0<x<1&\\&-1&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}$	${\displaystyle {\frac {1}{s}}}$	${\displaystyle -\infty <\Re s<0}$
${\displaystyle u(1-x)\,x^{a}=\left\{{\begin{aligned}&x^{a}&&\;{\text{if}}\;0<x<1&\\&0&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}$	${\displaystyle {\frac {1}{s+a}}}$	${\displaystyle -\Re a<\Re s<\infty }$
${\displaystyle -u(x-1)\,x^{a}=\left\{{\begin{aligned}&0&&\;{\text{if}}\;0<x<1&\\&-x^{a}&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}$	${\displaystyle {\frac {1}{s+a}}}$	${\displaystyle -\infty <\Re s<-\Re a}$
${\displaystyle u(1-x)\,x^{a}\ln x=\left\{{\begin{aligned}&x^{a}\ln x&&\;{\text{if}}\;0<x<1&\\&0&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}$	${\displaystyle {\frac {1}{(s+a)^{2}}}}$	${\displaystyle -\Re a<\Re s<\infty }$
${\displaystyle -u(x-1)\,x^{a}\ln x=\left\{{\begin{aligned}&0&&\;{\text{if}}\;0<x<1&\\&-x^{a}\ln x&&\;{\text{if}}\;1<x<\infty &\end{aligned}}\right.}$	${\displaystyle {\frac {1}{(s+a)^{2}}}}$	${\displaystyle -\infty <\Re s<-\Re a}$
${\displaystyle {\frac {1}{1+x}}}$	${\displaystyle {\frac {\pi }{\sin(\pi s)}}}$	${\displaystyle 0<\Re s<1}$
${\displaystyle {\frac {1}{1-x}}}$	${\displaystyle {\frac {\pi }{\tan(\pi s)}}}$	${\displaystyle 0<\Re s<1}$
${\displaystyle {\frac {1}{1+x^{2}}}}$	${\displaystyle {\frac {\pi }{2\sin({\tfrac {1}{2}}\pi s)}}}$	${\displaystyle 0<\Re s<2}$
${\displaystyle \ln(1+x)}$	${\displaystyle {\frac {\pi }{s\,\sin(\pi s)}}}$	${\displaystyle -1<\Re s<0}$
${\displaystyle \sin(x)}$	${\displaystyle \sin({\tfrac {1}{2}}\pi s)\,\Gamma (s)}$	${\displaystyle -1<\Re s<1}$
${\displaystyle \cos(x)}$	${\displaystyle \cos({\tfrac {1}{2}}\pi s)\,\Gamma (s)}$	${\displaystyle 0<\Re s<1}$
${\displaystyle e^{ix}}$	${\displaystyle e^{i\pi s/2}\,\Gamma (s)}$	${\displaystyle 0<\Re s<1}$
${\displaystyle J_{0}(x)}$	${\displaystyle {\frac {2^{s-1}}{\pi }}\,\sin(\pi s/2)\,\left[\Gamma (s/2)\right]^{2}}$	${\displaystyle 0<\Re s<{\tfrac {3}{2}}}$	${\displaystyle J_{0}(x)}$是第一类贝塞尔函数。
${\displaystyle Y_{0}(x)}$	${\displaystyle -{\frac {2^{s-1}}{\pi }}\,\cos(\pi s/2)\,\left[\Gamma (s/2)\right]^{2}}$	${\displaystyle 0<\Re s<{\tfrac {3}{2}}}$	${\displaystyle Y_{0}(x)}$是第二类贝塞尔函数。
${\displaystyle K_{0}(x)}$	${\displaystyle 2^{s-2}\,\left[\Gamma (s/2)\right]^{2}}$	${\displaystyle 0<\Re s<\infty }$	${\displaystyle K_{0}(x)}$是第二类修正贝塞尔函数。

• Galambos, Janos; Simonelli, Italo. . Marcel Dekker, Inc. 2004. ISBN 0-8247-5402-6.