Fourier Transform formulas
Fourier Series formulas
∞
X
f (x) = a0 +
(ak cos(kx) + bk sin(kx))
Z k=1
π
1
f (x) dx
2π −π
Z
1 π
ak =
cos(kx)f (x) dx = ck + ck
π −π
Z π
1
bk =
sin(kx)f (x) dx = i(ck − c−k )
π −π
∞
X
f (x) =
ck eikx
a0 =
−∞
1
ck =
2π
square wave
f (x)
Z
π
f (x)e−ikx dx
−π
Fourier series
←→
∞
X
4
sin(kx)
πk
k=1,3,5,...
∞
X
Fourier series
←→
ck eikx
−∞
Z ∞
Fourier transform 1
2π
fˆ(ω)eiωx where ω =
2π −∞
T
Z ∞
−ikx
1
1
Fourier series
û(k) =
u(x)e
dx
δ(x)
←→
+ (cos x + cos 2x + cos 3x + .
−∞
2π
Z ∞
π
1
sin x sin 2x sin 3x
Fourier series
ikx
u(x) =
û(k)e dk
−
+
− ...
x
←→
2
2π −∞
1
2
3
sin x sin 2x sin 3x
Fourier series
û(k) = Ĝ(k)ĥ(k)
x
←→
2
−
+
−
.
.
.
Z ∞
1
2
3

u(x) = g(x)h(x) ⇐⇒ ĝ(k) ∗ ĥ(k) = û(k) =
ĝ(k − τ )ĥ(τ ) dτ
2


π
a0 = (
Z−∞
∞
Fourier series
0,
k odd.
←→
û(k) = ĝ(k)ĥ(k) ⇐⇒ g(x) ∗ h(x) = u(x) =
g(x − τ )h(τ ) dτ | sin(x)|

ak = 4 1 

−∞
π 1−k2 , k even.
Z ∞
eixd u(x) ⇐⇒ û(k − d)
F −1
ˆ
Ĥ(k) = f (k)ĝ(k) ←→ H(x) = f (x) ~ g(x) =
f (τ )g(x − τ
u(x − a) ⇐⇒ e−iak û(k)
f (x)
←→
−inf ty
du
⇐⇒ ikû(k)
dx
Z x
û(k)
u(x) dx ⇐⇒
+ cδ(k)
ik
a
u(x) = f (x) ~ g(x)
û(k) = fˆ(k)ĝ(k)
1
8. trig identities
1 1
− cos(2x)
2 2
1 1
cos2 (x) = + cos(2x)
2 2
3
1
3
sin (x) = sin(x) − sin(3x)
4
4
1
3
cos3 (x) = cos(x) − cos(2x)
4
2
sin(2x) = 2 sin(x) cos(x)
sin2 (x) =
cos(2x) = cos2 (x) − sin2 (x)
= 1 − 2 sin2 (x)
= 2 cos2 (x) − 1
2 tan(x)
tan(2x) =
1 − tan2 (x)
sin(A ± B) = sin(A) cos(B) ± cos(A) cos(B)
1.
2.
3.
4.
5.
cos(A ± B) = cos(A) sin(B) ∓ sin(A) sin(B)
Z
When period T is not 2π replace k by
in all
cosn−1 (x) sin(x) n − 1
n
cos (x) dx =
+
cosn−2 dx
formulas for Fourier series.
n
n
1
x
= cos x sin x +
n even
2
2
R∞
1
2
= cos2 x sin x + sin x n odd
Plancherel formula 2π −∞ |f (x)|2 dx
=
3
3
R∞
Z
Z
ˆ 2
n−1
−∞ |f (k)| dk
−
sin
(x)
cos(x)
n−1
n
sin (x) dx =
+
sinn−2 dx
n
n
−1
x
=
sin x cos x +
n even
2
2
∞
Rπ
P
2
−1
Parseval’s formula −π |f (x)|2 dx = 2π
|ck |2
sin2 x cos x − cos x n odd
=
k=1
3
Z
Z 3
1
n ax
n ax
n−1 ax
x e dx =
x e −n x
e dx
a
2
2
2
2
2
Parseval’s
=
R ∞ 2 formula again 2πa0 +π a1 + b1 + a2 + b2 + .9.. . exp/trig
f
(x)
dx
−∞
eiθ − e−iθ
sin(x) =
2i
eiθ + e−iθ
R∞
cos(x) =
Inner
products
2π −∞ f (x)ḡ(x) dx
=
2
R∞
ˆ ¯
reiθ = r (cos(θ) + i sin(θ))
−∞ f (k)ĝ(k) dk
2π
T k
Z
ln(reiθ ) = ln(r) + iθ + 2kπi
Z ∞
√
x2
x2
k2
F (e− 2 ) =
e− 2 e−ikx dx = e− 2 2π
R
R
6. integration by parts uv 0 = [uv] − u0 v so pick
the one that is easy to differentiate for u and the
one that is easy to integrate for v.
−∞
−x2
k2 √
) = e− 4
√
Z ∞
π
−x2
e
dx =
2
Z 0∞
√
2
e−x dx = π
F (e
7. properties of odd and even functions Let o, e be
odd and even functions, then e + e = e, o + o =
o, e × e = e, o × o = e, o × e = o, ee = e, oe = o
−∞
Laplace
2
π
1. To find solution to Laplace on disk, or radius r,
use polar. The solution is
If we are given u0 = δ at point on circle as boundary conditions, use the above formula, much easier.
misc. items
u(r, θ) = a0 + a1 r cos(θ) + b1 r sin(θ)
2
2
+ a2 r cos(2θ) + b2 r sin(2θ) . . .
Rb
1. The function that minimizes a 12 (u0 (x))2 −
f u(x) dx is the solution of u00 (x) = f
Where the ak and bk found from finding Fourier
series of u(r, θ) evaluated at boundary as normally
done.
2. every function is made up of odd/even parts
f (x) − f (−x)
2
f (x) + f (−x)
=
2
fodd part =
2. solution inside the circle is
Z π
1 − r2
1
=
u(r, θ) =
2
2π
−π 1 + r − 2r cos(θ − ζ) dζ
feven part
references:
1. schaum’s mathematical handbook of formulas and tables by Spiegel
2. http://www.integraltec.com/math
3. http://en.wikipedia.org/wiki/Integration_by_reduction_formulae
3