For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. We have the relation $(\mathcal{R})$: $\beta ( x , y ) = \frac { \Gamma ( x ) \Gamma ( y ) } { \Gamma ( x + y ) }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$. From the relation $(\mathcal{R})$, justify that $\frac { \partial \beta } { \partial y }$ is defined on $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$. Establish that for all real $x > 0$ and $y > 0 , \frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.
For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. We have the relation $(\mathcal{R})$: $\beta ( x , y ) = \frac { \Gamma ( x ) \Gamma ( y ) } { \Gamma ( x + y ) }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$.
From the relation $(\mathcal{R})$, justify that $\frac { \partial \beta } { \partial y }$ is defined on $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$.
Establish that for all real $x > 0$ and $y > 0 , \frac { \partial \beta } { \partial y } ( x , y ) = \beta ( x , y ) ( \psi ( y ) - \psi ( x + y ) )$.