Integral Transforms Formula Sheet
January 3, 2018 | Author: grrrmo | Category: N/A
Short Description
Formula Sheet that summarizes properties of fourier transform, laplace transform, fourier sine transform, fourier cosine...
Description
Integral Transforms Formula Sheet created by Jaime X. Lopez Definition of the FT and Convolution Z ∞ 1 F {f (x)} = F (k) = √ e−ikx f (x) dx 2π −∞ Z ∞ 1 −1 √ F {F (k)} = f (x) = eikx F (k) dk 2π −∞ Z ∞ 1 f (x) ∗ g (x) = √ f (x − ξ) g (ξ) dξ 2π −∞
Parseval’s Relation Z ∞ Z ∞ f (x) g (x)dx = F (k) G (k)dk −∞ −∞ Z ∞ Z ∞ 2 2 |f (x)| dx = |F (k)| dk −∞
1 0
if x > a if x < a
H (x − a) = 1, a ≤ x ≤ b χ[a,b] (x) = 0, elswhere 1 if x > 0 sgn (x) = −1 if x < 0 Z ∞ f (ξ) δ (ξ − x) dξ = f (x)
(1) (2) (3) (4) (5) (6)
−∞
d f (x, t) = ikF (k, t) F dx 2 d F f (x, t) = −k 2 F (k, t) dx2 n d n F f (x, t) = (ik) F (k, t) dxn n dn d F f (x, t) = F (k, t) dtn dtn d F {xf (x)} = i F (k) dk dn F {xn f (x)} = in n F (k) dk
(17) (18)
F {f (x) ∗ g (x)} = F (k) G (k)
(19)
x exp (−a |x|) , a > 0
Let F (k) = F {f (x)} and G (k) = F {g (x)}
−∞
(16)
qF (k)
a 2 π k2 +a2
exp (−a |x|) , a > 0
Properties and Identities of the FT
F {F (x)} = f (−k) Z ∞ F (k) g (k) eikx dk = f (ξ) G (ξ − x) dξ
f ∗ (g ∗ h) = (f ∗ g) ∗ h (f + g) ∗ h = f ∗ h + g ∗ h √ f ∗ e = e ∗ f = f, e = 2πδ
f (x)
Z ∞ n 1 exp −nx2 = eikx dk δ (x) = lim n→∞ π 2π −∞ x2 =1 I (x) = lim exp − n→∞ 4n
∞
(15)
Let F (k) = F {f (x)}
r
Z
f ∗g =g∗f
Transform Pairs
−∞
F {f (x − a)} = e−ika F (k) k 1 F F {f (ax)} = |a| a n o F f (−x) = F {f (x)} F eiax f (x) = F (k − a)
(14)
−∞
Properties of Convolution
Generalized Functions
(13)
(7) (8)
|x| exp (−a |x|) , a > 0 exp −ax2 , a > 0 x exp −ax2 , a > 0 x2 exp −ax2 , a > 0 xn exp −ax2 , a > 0 a x2 +a2 , a ax x2 +a2 , a
>0 >0 H (x)
χ[−a,a] (x) δ (x) δ (x − a) 1 x xn exp (iax) x exp (iax) sin ax x
δ (x − a) + δ (x + a) (9) (10) (11) (12)
1 of 2
δ (x − a) − δ (x + a) sgn (x) sin (x) cos (x) sin ax2 cos ax2
q
−2aik
2
q π (k22+a22)
2
2 a −k π (k2 +a2 )2 2 √1 exp − k 2a 4a
−ik k2 exp − 4a (2a)3/2 2 k −k2 − 4a 5/2 exp (2a) 2 2 (−ik) exp − k (2a)(2n+1)/2 p π −a|k| 4a e p π2 −a|k| ike p π 8 1 δ (k) + iπk 2q 2 sin ak π k √1 2π √1 exp(−ika) 2π √
√2πδ (k) i√ 2πδ 0 (k) n i√ 2πδ (n) (k) (k − a) √ 2πδ n (n) 2πi δ (k − a) pπ χ (k) [−a,a] q2 2
qπ
cos (ak)
2 i sin (ak) πq 2 1
π ik pπ −i [δ (k + a) − δ (k − a)] pπ2 [δ (k − a) 2 + δ (k + a)] √1 2a √1 2a
sin
cos
k2
4a2
k 4a
−
−
π 4 π 4
Integral Transforms Formula Sheet
Definition of the FCT and FST r Z ∞ 2 Fc {f (x)} = Fc (k) = f (x) cos(kx)dx π 0 r Z ∞ 2 Fc (k) cos(kx)dk Fc−1 {Fc (k)} = f (x) = π 0 r Z ∞ 2 Fs {f (x)} = Fs (k) = f (x) sin(kx)dx π 0 r Z ∞ 2 −1 Fs (k) sin(kx)dk Fs {Fs (k)} = f (x) = π 0
Properties of the Laplace Transform L e−at f (t) = f¯(s + a) L {tn f (t)} = (−1)n f¯(n) (s) Z ∞ f (t) = f¯(s)ds L t s L {f (t − a)H(t − a)} = e−as f¯(s) 1 ¯ s L {f (at)} = f |a| a 0 ¯ L {f (t)} = sf (s) − f (0) L {f 00 (t)} = s2 f¯(s) − sf (0) − f 0 (0)
Convolution Theorems for FCT and FST Z ∞ 1 f (ξ) [g(x + ξ) + g(|x − ξ|)] dξ Fc−1 {Fc Gc } = √ 2π 0 Z ∞ 1 Fc−1 {Fs Gs } = √ f (ξ) [g(ξ + x) + g(ξ − x)] dξ 2π 0 Z ∞ 1 g(ξ) [f (ξ + x) − f (ξ − x)] dξ Fs−1 {Fs Gc } = √ 2π 0 Z ∞ 1 Fs−1 {Fc Gs } = √ f (ξ) [g(ξ + x) − g(ξ − x)] dξ 2π 0
n o L f (n) (t) = sn f¯(s) − sn−1 f (0) − sn−2 f 0 (0) − · · · − sf (n−2) (0) − f (n−1) (0) Z t f¯(s) L f (τ )dτ = s 0 Convolution for the Laplace Transform L−1 f¯(s)¯ g (s) = f (t) ∗ g(t) Z t f (t) ∗ g(t) = f (t − τ )g(τ )dτ
Properties of the FCT and FST k 1 , a>0 Fc a a k 1 , a>0 Fs {f (ax)} = Fs a a r 2 0 Fc {f (x)} = kFs (k) − f (0) π r 2 0 Fc {f 00 (x)} = −k 2 Fc (k) − f (0) π Fs {f 0 (x)} = −kFc (k) r 2 00 2 Fs {f (x)} = −k Fs (k) − f (0) π Fc {f (ax)} =
(20)
Laplace Convolution also possesses the properties of associativity, commutativity, and distributivity.
(21) Special Functions (22)
Z
(23)
xa−1 e−x dx,
Γ(a + 1) = aΓ(a),
(25)
a>0
Standard Laplace Transforms (26)
f (t) 1
(27) tn (28) sin(bt) (29) cos(bt)
Definition of the Laplace Transform Z ∞ ¯ L {f (t)} = f (s) = e−st f (t)dt, Re(s) > 0
sinh(at) cosh(at)
0 c+i∞ Z
est f¯(s)ds,
a>0
Γ(n + 1) = n!, n ∈ N Z x 2 2 e−α dα erf(x) = √ π 0
(24)
0
1 L−1 f¯(s) = f (t) = 2πi
∞
Γ(a) = 0
Parseval’s Relation for FCT and FST Z ∞ Z ∞ Fc (k)Gc (k)dk = f (x)g(x)dx 0 Z ∞ Z0 ∞ 2 2 |f (x)| dx |Fc (k)| dk = 0 0 Z ∞ Z ∞ Fs (k)Gs (k)dk = f (x)g(x)dx 0 Z0 ∞ Z ∞ 2 2 |Fs (k)| dk = |f (x)| dx 0
0
eat
c>0
c−i∞
2 of 2
f¯(s) 1 s n! sn+1 b 2 s + b2 s s2 + b2 a s2 − a2 s 2 s − a2 1 s−a
s0 0 0 0 0 0 0 0
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