The document defines and provides equations for various special functions including the beta and gamma functions, Bessel function, error function, complementary error function, Heaviside's unit step function, pulse function, sinusoidal pulse, rectangle function, gate function, Dirac delta function, signum function, saw tooth wave function, triangular wave function, half-wave and full-wave rectified sinusoidal functions, and square wave function. The author is N. B. Vyas from the Department of Mathematics at Atmiya Institute of Tech. and Science in Rajkot, India.

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