Euler Integrals: Beta Function and Gamma Function
Definition
Euler Integrals The following two integrals are referred to as Euler integrals.
- $(a)$ Euler integral of the first kind: Beta function $$ B(p,q)=\int_{0}^1 t^{p-1}(1-t)^{q-1}dt,\quad p>0,\quad q>0 $$
- $(b)$ Euler integral of the second kind: Gamma function $$ \Gamma (p) = \int_{0}^\infty t^{p-1}e^{-t}dt,\quad p>0 $$
Explanation
Euler Integral of the First Kind
1-1. Beta Function: If the gamma function is considered a generalization of the factorial, then the beta function can be seen as a generalization of the binomial coefficient. $$ \begin{pmatrix} n \\ k \end{pmatrix}=\frac{ 1 }{ (n+1)B(n-k+1,k+1) } $$
1-2. Properties of Beta Function $$ B(p,q)=B(q,p) $$ $$ B(p,q)=B(p+1,q)+B(p,q+1) $$ $$ B(p+1,q)=\frac{ p }{p+q}B(p,q),\quad B(p,q+1)=\frac{ q }{p+q }B(p,q) $$ $$ B(p,p)=\frac{ 1 }{ 2^{2p-1} }B(p,{\textstyle \frac{ 1 }{ 2 }}) $$
1-3. Various Representations of Beta Function
$$ B(p,q)=\int_{0}^{a}\left( \frac{ t }{ a }\right)^{p-1} \left( 1-\frac{ t}{a}\right)^{q-1}\frac{ 1 }{a }dt=\frac{ 1 }{ a^{p+q-1} }\int_{0}^{a}t^{p-1}(a-t)^{q-1}dt $$ By substituting $t \rightarrow \dfrac{ t }{ a }$ into the definition of the beta function, it can be obtained immediately. $$ B(p,q)=2\int_{0}^{\pi/2}(\sin\theta)^{2p-1} (\cos \theta )^{2q-1}d\theta $$ $$ B(p,q)=\int_{0}^{\infty} \frac{ t^{p-1} }{(1+t)^{p+q}}dt $$ $$ B(p,q)=\frac{ \Gamma (p) \Gamma (q) }{ \Gamma (p+q) } $$
Euler Integral of the Second Kind
2-1. Gamma Function: If the beta function is considered a generalization of the binomial coefficient, then the gamma function can be seen as a generalization of the factorial. $$ \Gamma (n)=(n-1)! $$
2-2. Properties of Gamma Function $$ \Gamma (p+1) = p \Gamma (p) $$ $$ \Gamma (p) \Gamma (1-p) = \frac{ \pi }{ \sin (\pi p) } $$ There are also several important formulas that include the gamma function. ▶eq15◀