Gamma and Beta Functions

Beta Function

• The beta function, also known as the Euler integral of the first kind, is a special function in mathematics.
• It is defined as:

∫[0,1] t^(x-1) (1-t)^(y-1) dt

• The beta function is symmetric, meaning B(x, y) = B(y, x)
• It is used in various areas of mathematics, such as probability theory, statistics, and calculus.

Properties of Beta Function

• B(x, y) = Γ(x) Γ(y) / Γ(x+y)
• B(x, y) = B(x, x+y-1)
• B(x, 1) = 1/x
• B(x, x) = 1/(2x-1)

Gamma Function

• The gamma function, also known as the Euler integral of the second kind, is an extension of the factorial function to real and complex numbers.
• It is defined as:

Γ(z) = ∫[0,∞) t^(z-1) e^(-t) dt

• The gamma function is analytic for all complex numbers except for the non-positive integers.
• It is used in various areas of mathematics, such as probability theory, statistics, and calculus.

Properties of Gamma Function

• Γ(z) = (z-1)!
• Γ(z+1) = z Γ(z)
• Γ(z) Γ(1-z) = π / sin(πz)
• Γ(1/2) = √π

Relationships between Beta and Gamma Functions

• The beta function can be expressed in terms of the gamma function: B(x, y) = Γ(x) Γ(y) / Γ(x+y)
• The gamma function can be expressed in terms of the beta function: Γ(z) = B(z, 1) / z

Beta Function

• Defined as: ∫[0,1] t^(x-1) (1-t)^(y-1) dt
• Symmetric: B(x, y) = B(y, x)
• Used in: probability theory, statistics, and calculus

Properties of Beta Function

• B(x, y) = Γ(x) Γ(y) / Γ(x+y)
• B(x, y) = B(x, x+y-1)
• B(x, 1) = 1/x
• B(x, x) = 1/(2x-1)

Gamma Function

• Defined as: Γ(z) = ∫[0,∞) t^(z-1) e^(-t) dt
• Extension of the factorial function to real and complex numbers
• Analytic for all complex numbers except for the non-positive integers
• Used in: probability theory, statistics, and calculus

Properties of Gamma Function

• Γ(z) = (z-1)!
• Γ(z+1) = z Γ(z)
• Γ(z) Γ(1-z) = π / sin(πz)
• Γ(1/2) = √π

Relationships between Beta and Gamma Functions

• B(x, y) = Γ(x) Γ(y) / Γ(x+y)
• Γ(z) = B(z, 1) / z