Gamma Functions: ∞

1 √

xp−1 e−x dx (p > 0) Γ(p + 1) = pΓ(p) Γ = π 2 0

(2n)! √ π

Γ(p)Γ(1 − p) = π (for positive integer n)

Γ(n + 1/2) = n sin(πp) 4 n! Z ∞

dn 1

Γ(2n) = √ 22n−1 Γ(n)Γ(n + 1/2) n [Γ(p)] = xp−1 e−x ln(x)n dx dp π 0 Z

Γ(p) =

Beta Functions: 1

Z

xp−1 (1 − x)q−1 dx (p > 0, q > 0) B(p, q) = B(q, p) 0 Z a 1 Γ(p)Γ(q)

B(p, q) = y p−1 (a − y)q−1 dy

B(p, q) = p+q−1 Γ(p + q) a 0 Z ∞ Z π/2

y p−1

(sin θ)2p−1 (cos θ)2q−1 dθ B(p, q) = dy B(p, q) = 2 (1 + y)p+q 0 0  

1 a!

B(n, n) = B(n, 1/2)/22n−1 B(n, m) = C(a,b) = nC(n + m − 1, m − 1) (a − b)!b!

B(p, q) =

Error Functions, Gaussian Distribution: Z x Z ∞

2 2 2

−t2 erf(x) = √ e dt erf(∞) = 1 erfc(x) = √ e−t dt = 1 − erf(x) π 0 π x    Z x 4 2 x5 t 2 x3 2 + − ··· |x|  1 1 − t + − · · · dt = √ erf(x) = √ x− 2! 3 5 · 2! π 0 π r     Z Z x 2 x x 2 ∞ −t2 /2

1 1 1 erfc √ = e dt P(−∞, x) = √ e−t /2 dt = + erf √ π x 2 2 2 2π −∞ 2  

Z x 2 1 x 1

P(0, x) = √ e−t /2 dt = erf √

erf(−x) = −erf(x) 2 2π 0 2 Stirling’s Formula & Method of Frobenius:

n −n

n! ∼ n e

X p

p −p 2πn Γ(p + 1) ∼ p e 2πp y = an xn+s n=0

Elliptic Integrals: Z

φ

F(k, φ) =

1 − k 2 sin2 φ

0

Z F(k, φ) = 0

dφ p

x

Z

E(k, φ) =

1 p (1 − x2 )(1 − k 2 x2 )

φ

q 1 − k 2 sin2 φ dφ (Legendre Forms)

0

Z

dx E(k, φ) =

0

x

r

1 − k 2 x2 dx (Jacobi Forms) 1 − x2

For Jacobi form: x = sin φ Complete Elliptic Integrals: φ(upper limit) → π/2 Legendre’s Equation & Leibniz’s Rule for Differentiation: n  k   n−k 

dn X d d n!

(1 − x )y − 2xy + l(l + 1)y = 0 n (uv) = u v dx dxk dxn−k k!(n − k)! 2

00

0

k=0

Legendre Polynomials: −1/2 `

1 d` 2 x − 1

Φ(x, h) = 1 − 2xh + h2 2` `! dx` ∞ X Φ(x, h) = P0 (x) + hP1 (x) + h2 P2 (x) + · · · = h` P` (x) P` (x) =

|h| < 1

`=0

`P` (x) = (2` − 1)xP`−1 (x) − (` − 1)P`−2 (x) xP0` (x) − P0`−1 (x) = `P` (x)

P0` (x) − xP0`−1 (x) = `P`−1 (x) (1 − x2 )P0` (x) = `P`−1 (x) − `xP` (x)

Z 1

P` (x)Pm (x)dx = 0 (` 6= m) (2` + 1)P` (x) = P0`+1 (x) − P0`−1 (x)

−1

Z

1

−1

Z

P` (x)f (x)dx = 0 If f (x) is a polynomial of degree < `

1

[P` (x)]2 dx =

−1

2 2` + 1

Associated Legendre Functions:  (1 − x2 )y 00 − 2xy 0 + `(` + 1) −



m m2

2 m/2 d y = 0 Pm P` (x) ` (x) = (1 − x ) 2 1−x dxm

Bessel Functions:

x2 y 00 + xy 0 + (x2 − p2 )y = 0 y = AJp (x) + BNp (x) ∞ X

 x 2n+p

cos(πp)Jp (x) − J−p (x) (−1)n

Np (x) = Yp (x) = Γ(n + 1)Γ(n + p + 1) 2 sin(πp) n=0 ∞  x 2n−p

X (−1)n

J−p (x) =

J−p (x) = (−1)p Jp (x) (for integer p) Γ(n + 1)Γ(n − p + 1) 2 n=0

d 

 d p 2p

[x Jp (x)] = xp Jp−1 (x)

x−p Jp (x) = −x−p Jp+1 (x) Jp−1 (x) + Jp+1 (x) = Jp (x) dx dx x

p p

Jp−1 (x) − Jp+1 (x) = 2J0p (x) J0p (x) = − Jp (x) + Jp−1 (x) = Jp (x) − Jp+1 (x) x x  Z 1 0 (if a 6= b) 2  0 xJp (ax)Jp (bx) = 2 2 1 1 1 (if a = b) 0 2 [Jp+1 (a)] = 2 [Jp−1 (a)] = 2 Jp (x)   2 2 2 1 − 2a 0 a −p c y 00 + y + (bcxc−1 )2 + y = 0 → y = xa Zp (bxc ) x x2 r r

π π

jn (x) = J(2n+1)/2 (x) yn (x) = Y(2n+1)/2 (x) 2x 2x

Jp (x) =

Hermite Eq. & Laguerre Eq’s: y 00 − 2xy 0 + 2ny = 0 (Hermite)

n o

xy 00 + (1 − x)y 0 + ny = 0 xy 00 + (k + 1 − x)y 0 + ny = 0 (Laguerre)

n k 

2 d 1 x dn −x2

n −x k k d Hn (x) = (−1)n ex e x e e L (x) = L (x) = (−1) Ln+k (x)

n n dxn n! dxn dxk Sturm-Liouville Equation: d [A(x)y 0 ] + [λB(x) + C(x)] y = 0 dx PDEs:

1 ∂u 1 ∂2u

∇2 u = 0 (Laplace) ∇2 u = f (x, y, z) (Poisson) ∇2 u = 2 (Heat flow) ∇2 u = 2 2 (Wave) α ∂t v ∂t

â&#x2C6;&#x2021;2 in Spherical & Cylindrical: 1 â&#x2C6;&#x201A; â&#x2C6;&#x2021; u= 2 r â&#x2C6;&#x201A;r 2

 r

2 â&#x2C6;&#x201A;u



â&#x2C6;&#x201A;r

1 â&#x2C6;&#x201A; + 2 r sin Î¸ â&#x2C6;&#x201A;Î¸



â&#x2C6;&#x201A;u sin Î¸ â&#x2C6;&#x201A;Î¸



  1 â&#x2C6;&#x201A; 1 â&#x2C6;&#x201A; 2 u

2 â&#x2C6;&#x201A;u 1 â&#x2C6;&#x201A;2u â&#x2C6;&#x201A;2u â&#x2C6;&#x2021; u = + 2 + 2 2 r +

r â&#x2C6;&#x201A;r â&#x2C6;&#x201A;r r2 â&#x2C6;&#x201A;Î¸2 â&#x2C6;&#x201A;z r sin Î¸ â&#x2C6;&#x201A;Ď&#x2020;2

Solutions to Laplaceâ&#x20AC;&#x2122;s Equation: 

Jn (kr) Nn (kr)

u=



sin nÎ¸ cos nÎ¸



ekz eâ&#x2C6;&#x2019;kz





(Cylindrical) u =

rl râ&#x2C6;&#x2019;lâ&#x2C6;&#x2019;1



Pm l (cos Î¸)



sin mĎ&#x2020; cos mĎ&#x2020;

 (Spherical)

Solutions to Wave Equation: 

sin Ď&#x2030;t

Î˝ = frequency(secâ&#x2C6;&#x2019;1 ) Ď&#x2030; = 2Ď&#x20AC;Î˝ = kv y= (Cartesian)

2Ď&#x20AC;Î˝ cos Ď&#x2030;t Îť = wavelength k = 2Ď&#x20AC; Îť = v =     Jn (kr) sin nÎ¸ sin kvt z= (Cylindrical) Nn (kr) cos nÎ¸ cos kvt 

sin kx cos kx



v = ÎťÎ˝ Ď&#x2030; v

Laplace Transforms: Z

â&#x2C6;&#x17E;

f (t)eâ&#x2C6;&#x2019;pt dt = F (p) L [f (t) + g(t)] = L(f ) + L(g) L [cf (t)] = cL(f ) 0

L (y 0 ) = pY â&#x2C6;&#x2019; y0 L (y 00 ) = p2 Y â&#x2C6;&#x2019; py0 â&#x2C6;&#x2019; y00

L(f ) =

Fourier Transforms: r Z â&#x2C6;&#x17E; Z â&#x2C6;&#x17E;

1 2

â&#x2C6;&#x2019;iÎąx f (x)e dx fs (x) = gs (Îą) sin(Îąx)dÎą f (x) = g(Îą)e dÎą g(Îą) = 2Ď&#x20AC; Ď&#x20AC; â&#x2C6;&#x2019;â&#x2C6;&#x17E; 0 â&#x2C6;&#x2019;â&#x2C6;&#x17E; r Z â&#x2C6;&#x17E; r Z â&#x2C6;&#x17E; r Z â&#x2C6;&#x17E;

2 2 2

gs (Îą) = fs (x) sin(Îąx)dx fc (x) = gc (Îą) cos(Îąx)dÎą gc (Îą) = fc (x) cos(Îąx)dx Ď&#x20AC; 0 Ď&#x20AC; 0 Ď&#x20AC; 0 Z

â&#x2C6;&#x17E;

iÎąx

Convolution; Parsevalâ&#x20AC;&#x2122;s Theorem: Z â&#x2C6;&#x17E; 1 |f (x)|2 dx gâ&#x2C6;&#x2014;hâ&#x2030;Ą |g(Îą)| dÎą = 2Ď&#x20AC; â&#x2C6;&#x2019;â&#x2C6;&#x17E; 0 â&#x2C6;&#x2019;â&#x2C6;&#x17E;

1

g1 Âˇ g2 and f1 â&#x2C6;&#x2014; f2 are a pair of Fourier transforms g1 â&#x2C6;&#x2014; g2 and f1 Âˇ f2 are too 2Ď&#x20AC; Z

t

Z

g(t â&#x2C6;&#x2019; Ď&#x201E; )h(t)dĎ&#x201E; G(p)H(p) = L (g â&#x2C6;&#x2014; h)

â&#x2C6;&#x17E;

2

Green Functions: Z â&#x2C6;&#x17E;

d2

0 2 0 0 f (t )Î´(t â&#x2C6;&#x2019; t)dt 2 G(t, t ) + Ď&#x2030; G(t, t ) = Î´(t â&#x2C6;&#x2019; t) y(t) = G(t, t0 )f (t0 )dt0 dt 0 0 If homogenous solutions are known of y 00 + p(x)y 0 + q(x)y = f (x) then Z Z y1 (x)f (x) y2 (x)f (x) yp = y2 (x) dx â&#x2C6;&#x2019; y1 (x) dx y1 (x)y20 (x) â&#x2C6;&#x2019; y10 (x)y2 (x) y1 (x)y20 (x) â&#x2C6;&#x2019; y10 (x)y2 (x)

Z

y + Ď&#x2030; y = f (t) f (t) = 00

2

â&#x2C6;&#x17E;

0

0

0

Where y1 (x), y2 (x) are solutions of the homogenous eq. with y1 (a) = y2 (b) = 0.

Laplace Transforms: y = f (t) 1 sin at tk , k > −1 e−at −e−bt b−a

sinh at t sin at e sin bt −at

1 − cos at sin at − at cos at sin at t e−at −e−bt t

J0 (at)

Y = L(y) = F (p) 1 p a p2 +a2 Γ(k+1) pk+1 1 (p+a)(p+b) a p2 −a2 2ap (p2 +a2 )2 b (p+a)2 +b2 a2 p(p2 +a2 ) 2a3 (p2 +a2)2  arctan ap   p+b ln p+a 2 2 −1/2

(p + a ) −ap

−bp

y = f (t) e−at cos at k −at t e , k > −1 ae−at −be−bt a−b

cosh at t cos at e cos bt −at

at − sin at e−at (1 − at) sin at cos bt  t 

1 − erf

a √ 2 t

,a > 0

u(t − a)

e −e u(t − a) − u(t − b) δ(t − a) p −pa g(t − a)u(t − a) e G(p) e−at g(t)  g(t) p 1 g(at), a > 0 aG a t R R n t t d tn g(t) (−1)n dp g(t − τ )h(t)dτ = g(t)h(t − τ )dτ n G(p) 0 0 Rt 1 g(τ )dτ p G(p) 0  1, t > a > 0 u(t − a) = (Unit step function) 0, t < a

Y = L(y) = F (p) 1 p+a p p2 +a2 Γ(k+1) (p+a)k+1 1 (p+a)(p+b) p p2 −a2 p2 −a2 (p2 +a2 )2 p+a (p+a)2 +b2 a3 p2 (p2 +a2 ) p  2     (p+a) + arctan a−b 1/2 arctan a+b p p √ 1 −a p pe 1 −pa pe −pa

e G(p + a) R∞ G(u)du p G(p)H(p)

Cheat sheet

Whole lotta symbols