Class notes
Integral Transforms 2022/23
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Introduction to Fourier and Laplace transforms and their reverse transform and their use to solve partial differential equations
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IntegraltranformsSo we have focused on solving PDEs on boundeddomains Integral transforms can
far
help solvePDEs on unbounded domains
Integral transforms are operators
of the form
I fix ca JexMexRdx
for some function Mix k and constraints a b
FOURIER TRANSFORMS
Consider Fourier series
of a function fix on EL L
Aim take L to
fix Egancosknxtbusinknx withkn ng
I one with
café II Exercise
So
with Cn
II fix e ax Verify
f x
I É Igwetimeeiknx
Note fix Éckna kn fix e d ein
I kntz
n
It
In the limit L x thishas the form
of a Riemannintegral Skg kn fgadk
So in limit L a
dolt
f fyi
fix e e dk FOURIER INTEGRAL
FORMULA
Wedefine the FOURIERTRANSFORM to be
InLte eimax
F
fixD FCK
Integral formula produces INVERSETRANSFORM
F Jca fax jade elk
, Beware 1 Differentconventions exist wrt placementoffactor of21T
2 Differentnotationisused e
g fad fad FCK
3 There arevariousconvergence assumptionin this derivation
It turnsout that onsufficientconditionforexistenceoffac is fitfulaxe to
CALCULATING FOURIERTRANSFORMS
EXAMPLE
Find Jad if fix I 1 1 Kk s
Kbs
Fck q Ja txt feikdx fxeimdxt.fxe
imd I .EC
e ax
Ig
e
I IE
if e
EE on
E É't le
Yi Lege
e 4
7,41
4
E É e face
Tiki
Winnie
Note FT of EVENfunctions are always REAL
FT ODD function are always PURE IMAGINARY
of
because
FCK
I fWqyx ifeng.gsax and flood 0
Note it is typical that functions localisedinspace x become nonlocal in wavenumberspec
G and viceversa
e
g F six a try six de ax 1 Eff coskagginke
hd
Sfcw 81ktw
I Ife
FCsinwx Exercise Cuse Six a dp
Ig
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