complex form fourier integral

FOURIER SERIES AND INTEGRALS 4 1 FOURIER SERIES FOR PERIODIC FUNCTIONS This section explains three Fourier series: sines cosines and exponentials eikx Square waves (1 or 0 or −1) are great examples with delta functions in the derivative We look at a spike a step function and a ramp—and smoother functions too Start with sin x

Complex form The Fourier integral (3) also possesses an equivalent complex form or exponential form: f(x) = 1 π ∞ 0 ∞ −∞ f(t)[cosαt cosαx + sinαt 

Fourier Series to Fourier Integral Fourier Cosine and Sine Series Integrals The Complex Form of Fourier Integral Formula of Fourier Integral The Fourier Integral of f(x) de ned on the interval (1 1 Z 1 f(x) = A( ) cos( x) d ; 1) is given by Z 1 B( ) sin( x) d ; (1) 0 0 where Z 1 A( ) = f(t) cos( t) dt; 1 and Z 1 B( ) = f(t) sin( t) dt: 1

Fourier Integral The Complex Form of Fourier Integral The Complex Form of Fourier Integral The complex form of Fourier integral is given by f(x) = 1 2π

Fourier integral i e ( T)=∫[ ( S) K O S T+ ( S)???? T] ∞ 0 Where A(w)=1 ???? ∫ ( R) R ∞ −∞ &B(w)=∫ ( R)???? R ∞ −∞ This representation is known as the Fourier integral for the function f(x) NOTE: If f(x) is not continuous at x=a then ( +0)+ ( −0) 2 (where + r)=lim ℎ→0

1 Derivation of the complex form Begin with a real periodic function F(t) with period T: F(t + T) = F(t): We are going to write this as a series in complex exponentials F(t) = a0 + a1ei!t + a2e2i!t + + aneni!t + a 1e i!t + a 2e 2i!t + a ne ni!t + using both positive and negative indices so we can write n=+1 X F(t) = anein!t: n=1 Here ! = 2 =T