Beta function

Beta function is defined as below, for :

Aka. Eulerian integral of the first kind.

Note

For , the beta function is divergent.

Properties

Symmetrical

From the definition:

Proof Hint

Use .

Relation with gamma function

.

Transformations

This section is intended to be exam-focused. Proofs for the transformations are included in a separate section.

Form 0, 6

Form 0 (definition) is derived by setting and , .

Form 1, 3

Form 1 is derived by setting .

Form 2

Form 4

Form 5, 7

Form 5 is derived by setting .

Transformations Proofs

Form 1

Proof Hint

Use in the definition.

Form 2

Proof Hint

Use in Form 1.

Form 3

Proof Hint

Use in Form 1.

Form 4

Proof Hint

Use in Form 3.

Form 5

Proof Hint

Use the substituition in the definition.

Form 6

Proof Hint

Use in the definition.

Form 7