table of fourier transforms

School of Mathematical Sciences ENG2092 Fourier transform table ENG2092 ADVANCED ENGINEERING MATHEMATICS B Complex Fourier series expansion A periodic function () f t with period can often be expressed in terms of the series 0 T 0 () n n in t ft ce ω ∞ =−∞ = ∑ where 0 2/ T 0 ω π = and the complex coefficients are given by n c 0 0 0 0 1 () n T in t c fte T ω − = ∫ dt . A table of some Fourier transforms () f t ˆ ( ) ( ) () i t f f fte ω ω ∞ −∞ − = = ∫ F dt 1 1 1 ˆ ˆ ( ) ( ) 2 i t f f e d ω ω ω π ∞ − −∞ = ∫ F ˆ ( ) f ω 2 ( ) ( ut a ut a + − − ) (2sin )/ aω ω 3 () t e ut α − for 0 α > /( ) i i ω α − − 4 ( ) t e u t α − for 0 α > /( ) i i ω α + 5 t e α − for 0 α > 2 2 2 /( ) α ω α + 6 sgn( ) t te α − for 0 α > 2 2 2 /( i ) ω ω α − + 7 2 2 /( ) t α α + exp( ) π αω − 8 ( ) t a δ − i a e ω − 9 '( ) f t ˆ ( ) i f ω ω 10 () i t e ft α ˆ ( ) f ω α − 11 () tft ˆ '( ) if ω 12 ( ) f t a − ˆ ( ) i a e f ω ω −

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Table with details about Fourier Transforms

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School of Mathematical SciencesENG2092

Fourier transform table

ENG2092 ADVANCED ENGINEERING MATHEMATICS B

Complex Fourier series expansion

A periodic function ( )f t with period can often be expressed in terms of the series 0T

0( ) nnin tf t c e ==

where 0 2 /T0 = and the complex coefficients are given by nc0 0

00

1 ( )nT in tc f t e

T= dt .

A table of some Fourier transforms

( )f t ( ) ( ) ( ) i tf f f t e = = F dt 1 1

1 ( ) ( )2

i tf f e d = F ( )f

2 ( ) (u t a u t a+ ) (2sin ) /a 3 ( )te u t for 0 > / ( )i i 4 ( )te u t for 0 > / ( )i i + 5 te for 0 > 2 22 / ( ) + 6 sgn( ) tt e for 0 > 2 22 / (i ) + 7 2 2/ ( )t + exp( ) 8 ( )t a i ae 9 '( )f t ( )i f

10 ( )i te f t ( )f 11 ( )t f t '( )i f 12 ( )f t a ( )i ae f

ENG2092 ADVANCED ENGINEERING MATHEMATICS BComplex Fourier series expansionA table of some Fourier transforms