The document defines the beta and gamma functions, explores their properties, and examines the relationship between them. The beta function is defined as the integral of x^(p-1)(1-x)^(q-1) from 0 to 1, where p and q are greater than 0. The gamma function is defined as the integral of x^(α-1)e^(-x) from 0 to infinity. A key relationship expressed is that the beta function equals the gamma function of m times the gamma function of n, divided by the gamma function of m + n. Applications of the beta and gamma functions include areas like probability, statistics, and electromagnetism.

1. Beta Gamma Functions
and the Relation Between
Beta and Gamma:
N.Tanvi
23R01A05B1
CSE-B

2. Definition of Beta Function
Beta Functions :
The beta function formula is defined as follows:
where p>0 and q>0

3. Properties of Beta Function
Symmetric properties:
for p> 0 and q > 0

4. Definition of Gamma Function
Gamma Function:
The gamma function then is defined as

5. Properties of Gamma Function:
• Γ(α +1)=α Γ(α)
• Γ(m)=(m-1)!, for m = 1,2,3 …;
• Γ(½) = √π

6. Relation Between Beta and Gamma
Functions:
The relationship between beta and gamma function can be mathematically expressed as-
Β (m,n) = ΓmΓn/ Γ (m+n)

7. Example of Beta and Gamma Functions:

8. Probability
Statistics
Complex analysis
Number Theory
Quantum mechanism
Electromagnetism
Applications of Beta and Gamma functions:

9. THANK YOU