Dr. Nirav Vyas Special Function.pdf
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Transcript of 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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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)
N. B. Vyas, AITS - Rajkot Special Functions
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)
N. B. Vyas, AITS - Rajkot Special Functions
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.
N. B. Vyas, AITS - Rajkot Special Functions
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.
N. B. Vyas, AITS - Rajkot Special Functions
Pulse Unit Height & Duration T
The pulse of unit height and duration T is defined by
f(t) =
{1 , 0 < t < T0 , T < t
N. B. Vyas, AITS - Rajkot Special Functions
Sinusoidal Pulse
The sinusoidal pulse is defined by
f(t) =
{sinat , 0 < t < π
a0 , πa < t
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
Gate Function
The gate function is defined as
fa(t) =
{1 , |t| < a0 , |t| > a
N. B. Vyas, AITS - Rajkot Special Functions
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.
N. B. Vyas, AITS - Rajkot Special Functions
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.
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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)
N. B. Vyas, AITS - Rajkot Special Functions
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)
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions
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
N. B. Vyas, AITS - Rajkot Special Functions