Introduction to some special function Unit -1, Advanced Engineering Mathematics(130002), GTU

1. Special Functions
N. B. Vyas
Department of Mathematics,
Atmiya Institute of Tech. and Science,
Rajkot-360005 (Guj.), INDIA.
niravbvyas@gmail.com
N. B. Vyas, AITS - Rajkot Special Functions

2. Special Functions
1 Beta & Gamma functions
2 Bessel function
3 Error function & Complementary error function
4 Heaviside’s Unit Step Function
5 Pulse Unit Height & Duration
6 Sinusoidal pulse
7 Rectangle function
8 Gate function
9 Dirac Delta function
10 Signum function
11 Saw tooth wave function
12 Triangular wave function
13 Half-wave Rectified Sinusoidal function
14 Full-wave Rectified Sinusoidal function
15 Square wave function
N. B. Vyas, AITS - Rajkot Special Functions

3. Error function & Complementary error function
The error function is defined by the integral
erf(z) =
2
√
π
z
0
e−t2
dt , z may be real or complex variable.
N. B. Vyas, AITS - Rajkot Special Functions

4. Error function & Complementary error function
The error function is defined by the integral
erf(z) =
2
√
π
z
0
e−t2
dt , z may be real or complex variable.
This function appears in probability theory, heat conduction theory
and mathematical physics.
When z = 0 ⇒ erf(0) = 0 and
erf(∞) =
2
√
π
∞
0
e−t2
=
Γ(1
2)
√
π
= 1
N. B. Vyas, AITS - Rajkot Special Functions

5. Error function & Complementary error function
The complementary error function is defined by the integral
erfc(z) =
2
√
π
∞
z
e−t2
dt , z may be real or complex variable.
N. B. Vyas, AITS - Rajkot Special Functions

6. Error function & Complementary error function
The complementary error function is defined by the integral
erfc(z) =
2
√
π
∞
z
e−t2
dt , z may be real or complex variable.
Using the properties of integral , we note that
erfc(z) = 2√
π
∞
0
e−t2
dt − 2√
π
z
0
e−t2
dt
= 2√
π
√
π
2 − erf(z)
= 1 − erf(z)
N. B. Vyas, AITS - Rajkot Special Functions

7. Heaviside’s Unit Step Function
The Heaviside’s Unit Step function (also known as delayed unit
step function) is defined by
H(t − a) =
1 , t > a
0 , t < a
N. B. Vyas, AITS - Rajkot Special Functions

8. Heaviside’s Unit Step Function
The Heaviside’s Unit Step function (also known as delayed unit
step function) is defined by
H(t − a) =
1 , t > a
0 , t < a
It delays output until t = a and then assumes a constant value of 1
unit.
If a = 0 then
H(t) =
1 , t > 0
0 , t < 0
which is generally called as unit step function.
N. B. Vyas, AITS - Rajkot Special Functions

9. Pulse Unit Height & Duration T
The pulse of unit height and duration T is defined by
f(t) =
1 , 0 < t < T
0 , T < t
N. B. Vyas, AITS - Rajkot Special Functions

10. Sinusoidal Pulse
The sinusoidal pulse is defined by
f(t) =
sinat , 0 < t < π
a
0 , π
a < t
N. B. Vyas, AITS - Rajkot Special Functions

11. Rectangle Function
The rectangle function is defined by
f(t) =
1 , a < t < b
0 , otherwise
N. B. Vyas, AITS - Rajkot Special Functions

12. Rectangle Function
The rectangle function is defined by
f(t) =
1 , a < t < b
0 , otherwise
In term of Heaviside unit step function, we have
f(t) = H(t − a) − H(t − b)
If a = 0 , then rectangle reduces to pulse of unit height and duration b
N. B. Vyas, AITS - Rajkot Special Functions

13. Gate Function
The gate function is defined as
fa(t) =
1 , |t| < a
0 , |t| > a
N. B. Vyas, AITS - Rajkot Special Functions

14. Dirac Delta Function
Consider the function fε(t) defined by
fε(t) =
1
ε , 0 ≤ t ≤ ε
0 , t > ε
where ε > 0.
N. B. Vyas, AITS - Rajkot Special Functions

15. Dirac Delta Function
Consider the function fε(t) defined by
fε(t) =
1
ε , 0 ≤ t ≤ ε
0 , t > ε
where ε > 0.
we note that as ε → 0, the height of the rectangle increases
indefinitely and width decreases in such a way that its area is always
equal to 1.
N. B. Vyas, AITS - Rajkot Special Functions

16. Signum Function
The signum function , denoted by sgn(t) , is defined by
sgn(t) =
1 , t > 0
−1 , t < 0
N. B. Vyas, AITS - Rajkot Special Functions

17. 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 so
sgn(t) = 2H(t) − 1
N. B. Vyas, AITS - Rajkot Special Functions

18. Saw Tooth Wave Function
The saw tooth function f with period a is defined by
f(t) =
t , 0 ≤ t < a
0 , t ≤ 0
f(t + a) = f(t)
N. B. Vyas, AITS - Rajkot Special Functions

19. Saw Tooth Wave Function
The saw tooth function f with period a is defined by
f(t) =
t , 0 ≤ t < a
0 , t ≤ 0
f(t + a) = f(t)
The saw tooth function with period 2π is defined as
f(t) =
t , −π < t < π
0 , otherwise
N. B. Vyas, AITS - Rajkot Special Functions

20. Triangular Wave Function
The triangular wave function f with period 2a is defined by
f(t) =
t , 0 ≤ t < a
2a − t , a ≤ t < 2a
f(t + 2a) = f(t)
N. B. Vyas, AITS - Rajkot Special Functions

21. Half-Wave Rectified Sinusoidal Function
The half-wave rectified sinusoidal function f with period 2π is
defined by
f(t) =
sint , 0 < t < π
0 , π < t < 2π
f(t + 2π) = f(t)
N. B. Vyas, AITS - Rajkot Special Functions

22. 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 period
π
ω
N. B. Vyas, AITS - Rajkot Special Functions

23. 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)
N. B. Vyas, AITS - Rajkot Special Functions

24. N. B. Vyas, AITS - Rajkot Special Functions