Beta and Gamma Functions: Definition, Relationship, Properties & Applications

Beta and gamma functions are popular functions in mathematics. Gamma is a single variable function while beta is a dual variable function. Beta function is used for computing and representing scattering amplitude for Regge trajectories. Also, it is applied in calculus using related gamma functions. Gamma function is like a factorial for natural numbers, its extension to positive real numbers makes it useful for modeling situations with continuous change, differential equations, complex analysis, and statistics.

Read More: Differential Equations Applications

Key Terms: Beta, Gamma, Calculus, Gamma function formula, Value of gamma, Gamma formula, Differential Equations

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Beta Function 

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Beta function, also known as Euler integral of the first kind, is defined by the integral

For complex number inputs x, y such that Re (x )> 0, Re (y )> 0

It is a symmetric function for all inputs x and y given by,

B(x,y)= B(y,x)

Beta and Gamma Functions Video Explaination

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Uses of Beta Function 

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The extension of the classical beta function has many uses. It helps in providing new extensions of the beta distribution, providing new extensions of the Gauss hypergeometric functions and confluent hypergeometric function and generating relations, and extension of Riemann-Liouville derivatives.

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Gamma Function

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Gamma function is a commonly used extension of the factorial function to complex numbers. It is defined for all complex numbers except non-positive integers.

For complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:

Where Re(z) > 0

Since the gamma function has no zeros, the reciprocal gamma function is an entire function.

While other extensions of the factorial function do exist, the gamma function is the most popular. It has various applications in the fields of probability and statistics, as well as combinatorics.

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Uses of Gamma Function 

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Gamma function has important applications in calculus, differential equations, complex analysis, and statistics. While the gamma function behaves like a factorial in the case of natural numbers which is a discrete set, its extension to positive real numbers which is a continuous set, makes the gamma function useful for modeling situations involving continuous change.

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Relationship Between Beta and Gamma Functions 

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The relationship between beta and gamma function can be expressed as

β(m,n) = ΓmΓn/ Γ(m+n)

Where,

 β(m,n) is a beta function with two variables m and n.

Γm is a gamma function with a variable m.

Γn is a gamma function with variable n.

Beta & Gamma Function Relation

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Beta Function Properties

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Some of the important properties of beta functions are listed below:

• It is a symmetric function. Therefore, B(p,q)=B(q,p)
• B(p, q) = B(p, q+1) + B(p+1, q)
• B(p, q+1) = B(p, q). [q/(p+q)]
• B(p+1, q) = B(p, q). [p/(p+q)]
• B (p, q). B (p+q, 1-q) = π/ p sin (πq)

Read More: Analytic Function

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Incomplete Beta Functions

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When the beta function is written in its generalized form, it is known as the incomplete beta function.

It can be represented as follows:

It can be denoted as B_z(a,b).  When z=1, The incomplete beta function now becomes, B(1 : a, b) = B(a, b).

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Things to Remember

• Beta function, also known as Euler integral of the first kind, is defined for complex number inputs x, y such that Re (x )> 0, Re (y )> 0.
• Beta function is a symmetric function, i.e. βx,y= β(y,x) for all inputs x and y.
• Beta function helps in providing a new extension of the beta distribution, providing new extensions of the Gauss hypergeometric functions, confluent hypergeometric function, generating relations, and extension of Riemann-Liouville derivatives.
• While the gamma function behaves like a factorial in the case of natural numbers which is a discrete set, its extension to positive real numbers which is a continuous set, makes the gamma function useful for modeling situations involving continuous change.
• The relationship between beta and gamma function can be expressed as β(m,n) = ΓmΓn/ Γ(m+n)

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