
Function
We are all familiar with the factorial notation. is the product of first natural numbers. We also define . For all other values of , is undefined.
Consider the function:
This is a recurrence relation, but is defined only for integers.
The function is an extension of the factorial function, with its argument shifted down by . i.e.

Definition
For ,
It can also be defined as

Useful Points about Function
II)
III)
IV) For ,
V)

Solving Problems using Function
I) For problems containing a function of in the exponent, i.e. (NOTE THE NEGATIVE SIGN), substitute as , and use integration by substitution.
II) For problems containing an exponential function of , e.g. , substitute in such a way that we get in the numerator.
III) For problems involving , substitute as .
IV) For problems involving and functions, use the Euler’s identity,
V) Problems using the reduction formula

Function
function is another important function in mathematics. French mathematician Jacques Binet gave it the name. It is defined as
Alternatively,

Properties
I) The function is symmetric, i.e.
II) By substituting as , we get
III)
IV) Legendre’s duplication formula:

Solving Problems using Function
The problems are to be solved by using the definition of function, its relation to function etc.
For integrals of the kind , substitute as