dr. nirav vyas special function.pdf

Special Functions

N. B. Vyas

Department of Mathematics,Atmiya Institute of Tech. and Science,

Rajkot-360005 (Guj.), [email protected]

N. B. Vyas, AITS - Rajkot Special Functions

Special Functions

1 Beta & Gamma functions2 Bessel function3 Error function & Complementary error function4 Heaviside’s Unit Step Function5 Pulse Unit Height & Duration6 Sinusoidal pulse7 Rectangle function8 Gate function9 Dirac Delta function10 Signum function11 Saw tooth wave function12 Triangular wave function13 Half-wave Rectified Sinusoidal function14 Full-wave Rectified Sinusoidal function15 Square wave function


Error function & Complementary error function

The error function is defined by the integral

erf(z) =2√π

∫ z

0e−t

2dt , z may be real or complex variable.

This function appears in probability theory, heat conduction theoryand mathematical physics.When z = 0⇒ erf(0) = 0 and

erf(∞) =2√π

∫ ∞0

e−t2

=Γ(12)√π

= 1


Error function & Complementary error function

The complementary error function is defined by the integral

erfc(z) =2√π

∫ ∞z

e−t2dt , z may be real or complex variable.

Using the properties of integral , we note that

erfc(z) = 2√π

∞∫0

e−t2dt− 2√

π

z∫0

e−t2dt

= 2√π

(√π2

)− erf(z)

= 1− erf(z)


Heaviside’s Unit Step Function

The Heaviside’s Unit Step function (also known as delayed unitstep function) is defined by

H(t− a) =

{1 , t > a0 , t < a

It delays output until t = a and then assumes a constant value of 1unit.If a = 0 then

H(t) =

{1 , t > 00 , t < 0

which is generally called as unit step function.


Pulse Unit Height & Duration T

The pulse of unit height and duration T is defined by

f(t) =

{1 , 0 < t < T0 , T < t


Sinusoidal Pulse

The sinusoidal pulse is defined by

f(t) =

{sinat , 0 < t < π

a0 , πa < t


Rectangle Function

The rectangle function is defined by

f(t) =

{1 , a < t < b0 , otherwise

In term of Heaviside unit step function, we havef(t) = H(t− a)−H(t− b)If a = 0 , then rectangle reduces to pulse of unit height and duration b


Gate Function

The gate function is defined as

fa(t) =

{1 , |t| < a0 , |t| > a


Dirac Delta Function

Consider the function fε(t) defined by

fε(t) =

{1ε , 0 ≤ t ≤ ε0 , t > ε

where ε > 0.

we note that as ε→ 0, the height of the rectangle increasesindefinitely and width decreases in such a way that its area is alwaysequal to 1.


Signum Function

The signum function , denoted by sgn(t) , is defined by

sgn(t) =

{1 , t > 0−1 , t < 0

If H(t) is unit step function, then

H(t) =1

2[1 + sgn(t)]

and sosgn(t) = 2H(t)− 1


Saw Tooth Wave Function

The saw tooth function f with period a is defined by

f(t) =

{t , 0 ≤ t < a0 , t ≤ 0

f(t+ a) = f(t)

The saw tooth function with period 2π is defined as

f(t) =

{t ,−π < t < π0 , otherwise


Triangular Wave Function

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

f(t) =

{t , 0 ≤ t < a2a− t , a ≤ t < 2a

f(t+ 2a) = f(t)


Half-Wave Rectified Sinusoidal Function

The half-wave rectified sinusoidal function f with period 2π isdefined by

f(t) =

{sint , 0 < t < π0 , π < t < 2π

f(t+ 2π) = f(t)


Full Rectified Sine Wave Function

The full rectified sine wave function f with period π is defined by

f(t) =

{sint , 0 < t < π−sint , π < t < 2π

f(t+ π) = f(t) or by

f(t) = |sinωt| with periodπ

ω


Square Wave Function

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

f(t) =

{1 , 0 < t < a−1 , a < t < 2a

f(t+ 2a) = f(t)


dr. nirav vyas special function.pdf

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