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