Special Functions in Math

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics or other applications. In math, several functions are important enough to deserve their own names. Functions like algebraic and transcendental functions are elementary functions in math. The list of special functions in math is given below.

(Source: Wikipedia)

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List of special functions in math:

Basic special functions
Number theoretic functions
Hypergeometric and related functions
Riemann Zeta and related functions
Antiderivatives of elementary functions
Elliptic and related functions
Iterated exponential and related functions
Gamma and related functions
Bessel and related functions
Other standard special functions
Miscellaneous functions

The example problems of trigonometric integral and logarithmic integral function are given below. Both trigonometric integral and logarithmic integral function are antiderivatives of elementary functions.

Example problems of special function in math:

Example 1:

Evaluate the integral int log (1 + x2) dx

Solution:

By integrating by parts method, taking log (1 + x2) as the first function and 1 as the second function, we get

 int log (1 + x2) dx  = int (log (1 + x2) . 1) dx

The above equation can be written as

 int log (1 + x2) dx  = log (1 + x2) int 1 dx - int [d/dx {log (1 + x2)} * int  1 dx] dx

= log (1 + x2) * x - int  (2x)/(1+x2) x dx

= xlog (1 + x2) - 2int (1 - 1/(1+x2) dx

= xlog(1 + x2) - 2int dx + 2int (dx)/(1+x2)

= xlog(1 + x2) - 2x +2 tan-1x + C

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Example 2:

Integrate the trigonometric function sin-1(cos x).

Solution:

Step 1: Given function

sin-1 (cos x)

Step 2: Integrate the given function sin-1 (cos x) with respect to ' x ',

int sin-1 (cos x)dx = int sin-1{sin pi/2 - x} dx

= int (pi/2 - x)dx

= pi/2 int dx - int x dx

= (pix)/2 - x^2/2 + C