Fourier Series formula sheet



Fourier series is an expansion of a periodic function of period 2π2π which is representation of a function in a series of sine or cosine such as

f(x)=a0+n=1ancos(nx)+n=1bnsin(nx)f(x)=a0+n=1ancos(nx)+n=1bnsin(nx)

where a0a0 , anan and bnbn are constants and are known as fourier coefficients.
In applying fourier theorem for analysis of an complex periodic function , given function must satisfy following condition
(i) It should be single valued
(ii) It should be continuous.


 Drichlet’s Conditions(sufficient but not necessary)

When a function f(x)f(x) is to be expanded in the interval (a,b)
(a) f(a)f(a) is continous in interval (a,b) except for finite number of finite discontinuties.
(b) f(x)f(x) has finite number of maxima and minima in this interval.

Orthogonal property of sine and cosine functions
ππsin(mx)cos(mx)dx=0ππsin(mx)cos(mx)dx=0
ππsin(mx)sin(nx)dx=ππsin(mx)sin(nx)dx=[πδmnm00m=0]
ππcos(mx)cos(nx)dx=[πδmnm02πm=0]
Fourier Constants
a0=12πππf(x)dx
a0  is the average value of function f(x) over the interval
an=1πππf(x)cos(nx)dx
bn=1πππf(x)sin(nx)dx
For even functions
f(x)=f(x) and fourier series becomes
f(x)=a0+n=1ancos(nx)
 For odd functions
f(x)=f(x) and fourier series becomes
f(x)=a0+n=1ansin(nx)
Complex form of fourier series
putting c0=c0
cn=anibn2
and
cn=an+ibn2
f(x)=Cneinx
coefficent
Cn=12πππf(x)einxdx
Fourier series in interval (0,T)
General fourier series of a periodic piecewise continous function f(T) having period T=2πω is
f(t)=a0+n=1ancos(nx)+n=1bnsin(nx)
where
a0=1TT0f(t)dt
an=2TT0f(t)cos(nωT)dt
bn=2TT0f(t)sin(nωT)dt
 Complex Form of Fourier Series
f(x)=n=Cneiωt
where
cn=1TT0f(t)eiωtdx
 Advantages of Fourier series
1. It can also represent discontinous functions
2.  Even and odd functions are conveniently represented as cosine and sine series.
3.  Fourier expansion gives no assurance of its validity outside the interval.

Change of interval from (π,π) to (l,l)
Series will be
f(x)=a0+n=1ancos(nxπl)+n=1bnsin(nxπl)
with
a0=12lllf(x)dx
an=12lllf(x)cos(nπxl)dx
bn=12lllf(x)sin(nπxl)dx
Fourier Series in interval (0,l)
Cosine series when function f(x) is even
f(x)=a0+n=1ancos(nπxl)

Download Fourier Series formula sheet as PDF



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