Gamma function

Defined as below for :

Aka. Eulerian integral of the second kind.

Convergence

is convergent iff .

Proof Hint

Direct comparison test is used. Proved in 3 cases:

• Case 1: for positive integer

  • Consider the lemma 2’s limit definition
  • Take
  • Use the convergence of

• Case 2: for non-integers

  • Use
  • Use
  • is converging from case 1

• Case 3: for .

  • Proof is similar to case 1
  • But is an improper
  • Prove that it is also converging

Properties

Proofs are required for each property mentioned below.

• can be extrapolated from (see below for explanation)
• , where is a rational number (other than integers and half of any integer), cannot be expressed in a closed form value.

Extension of gamma function

function can be extended for negative non-integers using:

This cannot be used to define because of the denominator. And through induction, function cannot be defined for negative integers.

Lemmas

Lemma 1

Lemma 2

Transformations

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

Form 0, 1, 4

For :

Form 0 (definition) is resulted when setting . Form 1 is resulted when setting .

Form 2

Form 3

Transformations Proofs

Form 1

:

Proof Hint

Use .

Note

One of the most frequently used integrals in mathematics:

Form 2

Proof Hint

Use .

Form 3

Proof Hint

Use . If the given integral’s range is from to and there is , it’s better to try this substitution.

Form 4

For :

Proof Hint

Use .