ADVANCE ENGINEERING MATHEMATICS ( ) Branch : ME –D 1 Prepared by: NEHAL VAGHASIYA ( ) Guided By: Prof. TEJAS BHAVSAR INTRODUCTION TO SOME SPECIAL FUNCTIONS

KEY POINTS : Gamma function Beta function Bessel function Error functions Heaviside’s unit step solution The pulse of unit height and duration T Rectangle Function Gate Function Dirac Delta Function Signum Function Periodic function : (1) Saw Tooth Wave Function (2) Triangular Wave Function (3) Half-Wave Rectified Sinusoidal Function (4) Full Rectified Sine Wave Function (5) Square Wave Function

GAMMA FUNCTION The gamma function for is defined as

BETA FUNCTION The beta function B( x, y) is defined by Properties :

BESSEL FUNCTION Bessel’s functions play important role in applied mathematics and have many applicant in engineering. A Bessel function of order n is defined by

ERROR FUNCTIONS The complementary error function, denoted erfc, is defined as In mathematics, the error function (also called the Gauss error function) is a special function that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:

HEAVISIDE’S UNIT STEP SOLUTION  The Heaviside step function is a mathematical function denoted and known as the "unit step function."  The Heaviside's Unit Step function (also known as delayed unit step function) is defined by  If a = 0 then

THE PULSE OF UNIT HEIGHT AND DURATION T  The pulse of unit height and duration T is defined by SINUSOIDAL PULSE  The sinusoidal pulse is defined by

RECTANGLE FUNCTION The rectangle function is defined by In term of Heaviside unit step function, we have If a = 0, then rectangle reduces to pulse of unit height and duration b N. B.

GATE FUNCTION The gate function is defined as

DIRAC DELTA FUNCTION Consider the function defined by As the height of the rectangle increases indefinitely and width decreases in such a way that its area is always equal to 1.

SIGNUM FUNCTION The signum function, denoted by sgn(t), is defined by If H(t) is unit step function, then and so

PERIODIC FUNCTION Periodic function is a function that repeats its values in regular intervals or periods. A function f is said to be periodic with period P (P being a nonzero constant) if we have For example, the sine function is periodic with period 2π, since for all values of x. This function repeats on intervals of length 2π

SAW TOOTH WAVE FUNCTION The saw tooth function f with period a is defined by The saw tooth function with period 2 π is defined as

TRIANGULAR WAVE FUNCTION The triangular wave function f with period 2a is defined by

HALF-WAVE RECTIFIED SINUSOIDAL FUNCTION The half-wave rectified sinusoidal function f with period 2 π is defined by

FULL RECTIFIED SINE WAVE FUNCTION The full rectified sine wave function f with period π is defined by

SQUARE WAVE FUNCTION The square wave function f with period 2a is defined by