楊氏函數（Young's function）是一個以Γ函數定義的特殊函數[1]
C v ( z ) = ∑ n = 0 ∞ ( − 1 ) n z v + 2 n Γ ( v + 2 n + 1 ) {\displaystyle C_{v}(z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{v+2n}}{\Gamma (v+2n+1)}}}
C v ( z ) = x v Γ ( v + 1 ) ( 1 F 1 ( 1 ; v + 1 ; i z ) + 1 F 1 ( 1 ; v + 1 ; − i z ) {\displaystyle C_{v}(z)={\frac {x^{v}}{\Gamma (v+1)}}(_{1}F_{1}(1;v+1;iz)+_{1}F_{1}(1;v+1;-iz)}
C v ( z ) = a v ( 1 − a ( 1 / 2 − v ) ∗ v ∗ L o m m e l S 1 ( v + 1 / 2 , 3 / 2 , a ) v + 1 − a ( − v − 1 / 2 ) ∗ L o m m e l S 1 ( v + 3 / 2 , 1 / 2 , a ) v + 1 Γ ( v + 1 ) {\displaystyle C_{v}(z)={\frac {a^{v}(1-{\frac {a^{(}1/2-v)*v*LommelS1(v+1/2,3/2,a)}{v+1}}-{\frac {a^{(}-v-1/2)*LommelS1(v+3/2,1/2,a)}{v+1}}}{\Gamma (v+1)}}}
∫ 0 1 ( 1 − x ) v s i n ( a x ) d x = 1 z − Γ ( v + 1 ) z v + 1 C v ( z ) {\displaystyle \int _{0}^{1}(1-x)^{v}sin(ax)dx={\frac {1}{z}}-{\frac {\Gamma (v+1)}{z^{v+1}}}C_{v}(z)}
楊氏函數Maple動畫
