Integral Transforms Formula Sheet

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