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 .

Transformations

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

Proof Hint

I don’t know.

Relation with gamma function

.