Ppt of aem some special function
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AEM_2130002_SOME SPECIAL FUNCTIONS_IT/(A_1)_sem3
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Transcript of Ppt of aem some special function
AEM_2130002_SOME SPECIAL
FUNCTIONS_IT/(A_1)_sem3
Group members:
RITIKA AGARWAL
(130120116001)
NIRALI AKABARI
(130120116002)
AKANSHA SHARMA (130120116003)
UNIT STEP FUNCTION:
It is also known as heaviside unit step
function.
It is defined by :
PULSE OF UNIT HEIGHT:
Pulse of unit height of duration ‘t’ is
defined by
ax
axaxu
,0
,1)(
otherwise
Tttf
,0
0,1)(
SINUSOIDAL PULSE FUNCTION:
It is defined by
RECTANGLE FUNCTION:
A function is defined on arc is said to be rectangle function if
otherwiseo
axax
xf
,
0,sin)(
otherwise
bxaxf
,0
,1)(
)(xf
GATE FUNCTION
If function defined on arc is said to
be gate function if
DIRAC DELTA FUNCTION:
Let be a function defined on is said to be dirac delta function if
ax
axxf
||,0
||,1)(
otherwise
xxf
,0
0,1
)(
f r
)(xfa
SQUARE WAVE FUNCTION:
The function of period is defined
by
SAW TOOTH FUNCTION:
The function with period is
defined by
axa
axxf
2,1
0,1)(
)(xf
axxxf 0,)(
)(xf
a2
a2
TRIANGULAR WAVE FUNCTION:
The triangular wave function with
period
is defined by
FULL RECTIFIED SINE WAVE
FUNCTION:
The function is defined as with period
as:
axaxa
axxxf
2,2
0,)(
otherwise
tttf
,0
0,sin)(
)(tf
)(xf
a2
HALF WAVE RECTIFIED SINUSOIDAL
FUNCTION
It is defined as with period
PERIODIC FUNCTION
is said to be periodic if =
for all x ,if smallest positive no. of set of all
such are existing then that no. is called
fundamental period of .
2,0
0,sin)(
x
xxxf
)(xf
)(xf
)(xf
)(xf
2
)( axf
SIGNUM FUNCTION:
The signum function is defined by
0,1
0,1)(
x
xxf
f
•THE GAMMA FUNCTION:
If then the gamma function is
defined by
It is denoted by ┌n.
dxxe nx 1
0
0n
Properties of Gamma function
1.Reduction formula for gamma function
┌n+1=n ┌n, n>0.
2. If n is positive integer then ┌n+1=n!
3. Second form of gamma function
= ┌m
0
122
dxxe mx
2
1
4.Relation between gamma and beta
function:
=
5.┌
6.┌
2
1
nn
nn
4!
)!2()
2
1(
),( nmB
•THE BETA FUNCTION:
If then function is defined as
It is denoted by .
1
0
11 )1( dxxx nm
0,0 nm
),( nmB
Properties of Beta function
1. If the function is symmetric then
2.
3.
dxx
xnmB
nm
m
0
1
)1(),(
2
0
1212 cossin2),(
dnmB nm
),(),( mnBnmB
•ERROR FUNCTION:
The error function of is defined by
where may be real or complex
number.
It is denoted by .
dte
x
t
0
22
)(xerf
x
x
Properties of error function
1.
2.
3.
4. )()(
1)()(
1)(
0)0(
xerfxerf
xerfxerf
erf
erf
c
•THE COMPLIMENTARY ERROR
FUNCTION:
The complimentary error function is
defined by
It is denoted by .
x
t dte22
)(xerf c
