Engineering Mathematics Important Formulas [Latest ed.]

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ENGINEERING MATHEMATICS FORMULAS & SHORT NOTES HANDBOOK

Vector Algebra 2

If i, j, k are orthonormal vectors and A = A x i + A y j + A z k then | A| = A2x + A2y + A2z . [Orthonormal vectors ≡ orthogonal unit vectors.]

Scalar product A · B = | A| | B| cos θ





where θ is the angle between the vectors

Bx = A x Bx + A y B y + A z Bz = [ A x A y A z ]  B y  Bz

Scalar multiplication is commutative: A · B = B · A.

Equation of a line A point r ≡ ( x, y, z) lies on a line passing through a point a and parallel to vector b if r = a + λb

with λ a real number.

Page 1 of 24

Equation of a plane A point r ≡ ( x, y, z) is on a plane if either

(a) r · b d = |d|, where d is the normal from the origin to the plane, or y z x (b) + + = 1 where X, Y, Z are the intercepts on the axes. X Y Z

Vector product A × B = n | A| | B| sin θ, where θ is the angle between the vectors and n is a unit vector normal to the plane containing A and B in the direction for which A, B, n form a right-handed set of axes.

A × B in determinant form i j k Ax A y Az Bx B y Bz

A × B in matrix form    0 − Az A y Bx  Az 0 − Ax   By  − A y Ax 0 Bz

Vector multiplication is not commutative: A × B = − B × A.

Scalar triple product Ax A × B · C = A · B × C = Bx Cx

Ay By Cy

Vector triple product

A z Bz = − A × C · B, Cz

A × ( B × C ) = ( A · C ) B − ( A · B)C,

etc.

( A × B) × C = ( A · C ) B − ( B · C ) A

Non-orthogonal basis A = A1 e1 + A2 e2 + A3 e3 A1 = 0 · A

where 0 =

Similarly for A2 and A3 .

e2 × e3 e1 · (e2 × e3 )

Summation convention a

= ai ei

a·b

= ai bi

implies summation over i = 1 . . . 3 where ε123 = 1;

( a × b)i = εi jk a j bk εi jkεklm = δil δ jm − δimδ jl

Page 2 of 24

εi jk = −εik j

Matrix Algebra Unit matrices The unit matrix I of order n is a square matrix with all diagonal elements equal to one and all off-diagonal elements zero, i.e., ( I ) i j = δi j . If A is a square matrix of order n, then AI = I A = A. Also I = I −1 . I is sometimes written as In if the order needs to be stated explicitly.

Products If A is a (n × l ) matrix and B is a (l × m) then the product AB is defined by l

∑ Aik Bk j

( AB)i j =

k=1

In general AB 6= BA.

Transpose matrices If A is a matrix, then transpose matrix A T is such that ( A T )i j = ( A) ji .

Inverse matrices If A is a square matrix with non-zero determinant, then its inverse A −1 is such that AA−1 = A−1 A = I.

( A−1 )i j =

transpose of cofactor of A i j | A|

where the cofactor of A i j is (−1)i+ j times the determinant of the matrix A with the j-th row and i-th column deleted.

Determinants If A is a square matrix then the determinant of A, | A| (≡ det A) is defined by

| A| =

∑

i jk... A1i A2 j A3k . . .

i, j,k,...

where the number of the suffixes is equal to the order of the matrix.

2×2 matrices If A =



a c

b d



then,

| A| = ad − bc

AT =



a b

c d



A−1 =

1 | A|



d −c

−b a



Product rules ( AB . . . N ) T = N T . . . B T A T ( AB . . . N )−1 = N −1 . . . B−1 A−1

(if individual inverses exist)

| AB . . . N | = | A| | B| . . . | N |

(if individual matrices are square)

Orthogonal matrices An orthogonal matrix Q is a square matrix whose columns q i form a set of orthonormal vectors. For any orthogonal matrix Q, Q−1 = Q T ,

| Q| = ±1,

Q T is also orthogonal.

Page 3 of 24

Solving sets of linear simultaneous equations If A is square then Ax = b has a unique solution x = A −1 b if A−1 exists, i.e., if | A| 6= 0.

If A is square then Ax = 0 has a non-trivial solution if and only if | A| = 0. An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the error | Ax − b|) is the solution of the n equations A T Ax = A T b. If the columns of A are orthonormal vectors then x = A T b.

Hermitian matrices The Hermitian conjugate of A is A † = ( A∗ ) T , where A∗ is a matrix each of whose components is the complex conjugate of the corresponding components of A. If A = A † then A is called a Hermitian matrix.

Eigenvalues and eigenvectors The n eigenvalues λ i and eigenvectors u i of an n × n matrix A are the solutions of the equation Au = λ u. The eigenvalues are the zeros of the polynomial of degree n, Pn (λ ) = | A − λ I |. If A is Hermitian then the eigenvalues λi are real and the eigenvectors u i are mutually orthogonal. | A − λ I | = 0 is called the characteristic equation of the matrix A. Tr A = ∑ λi , i

also | A| =

∏ λi . i

If S is a symmetric matrix, Λ is the diagonal matrix whose diagonal elements are the eigenvalues of S, and U is the matrix whose columns are the normalized eigenvectors of A, then U T SU = Λ

and

S = UΛU T.

If x is an approximation to an eigenvector of A then x T Ax/( x T x) (Rayleigh’s quotient) is an approximation to the corresponding eigenvalue.

Commutators [ A, B] [ A, B] [ A, B]†

≡ AB − BA = −[ B, A]

= [ B† , A† ]

[ A + B, C ] = [ A, C ] + [ B, C ] [ AB, C ]

= A[ B, C ] + [ A, C ] B

[ A, [ B, C ]] + [ B, [C, A]] + [C, [ A, B]] = 0

Hermitian algebra b† = (b∗1 , b∗2 , . . .) Matrix form Hermiticity

b∗ · A · c = ( A · b)∗ · c

Eigenvalues, λ real

Au i = λ(i) ui

Orthogonality Completeness

Operator form Z

ψ∗ Oφ =

Bra-ket form

(Oψ)∗φ

ui · u j = 0 b = ∑ ui (ui · b)

φ = ∑ ψi

ψ∗i ψ j = 0

i

Z

ψ∗i φ

hψ|O|φi O |i i = λ i | i i

Oψi = λ(i)ψi Z

i

Z



hi | j i = 0 φ = ∑ |i i hi |φi i

Rayleigh–Ritz Lowest eigenvalue

b∗ · A · b λ0 ≤ b∗ · b

Z

λ0 ≤ Z

Page 4 of 24

ψ∗ Oψ ∗

ψ ψ

hψ|O|ψi hψ|ψi

(i 6 = j )

Pauli spin matrices σx =



0 1

 1 , 0

σ xσ y = iσ z ,

σy =



0 i

σ yσ z = iσ x ,

 −i , 0

σz =

σ zσ x = iσ y ,



1 0

0 −1



σ xσ x = σ yσ y = σ zσ z = I

Vector Calculus Notation φ is a scalar function of a set of position coordinates. In Cartesian coordinates φ = φ( x, y, z); in cylindrical polar coordinates φ = φ(ρ, ϕ, z); in spherical polar coordinates φ = φ(r, θ , ϕ); in cases with radial symmetry φ = φ(r). A is a vector function whose components are scalar functions of the position coordinates: in Cartesian coordinates A = iA x + jA y + kA z , where A x , A y , A z are independent functions of x, y, z.   ∂  ∂x     ∂  ∂ ∂ ∂   + j +k ≡ In Cartesian coordinates ∇ (‘del’) ≡ i   ∂y  ∂x ∂y ∂z    ∂  ∂z

grad φ = ∇φ,

div A = ∇ · A,

curl A = ∇ × A

Identities grad(φ1 + φ2 ) ≡ grad φ1 + grad φ2

grad(φ1φ2 ) ≡ φ1 grad φ2 + φ2 grad φ1

curl( A + A ) ≡ curl A1 + curl A2

div(φ A) ≡ φ div A + (grad φ) · A,

div( A1 + A2 ) ≡ div A1 + div A2

curl(φ A) ≡ φ curl A + (grad φ) × A

div( A1 × A2 ) ≡ A2 · curl A1 − A1 · curl A2

curl( A1 × A2 ) ≡ A1 div A2 − A2 div A1 + ( A2 · grad) A1 − ( A1 · grad) A2 div(curl A) ≡ 0,

curl(grad φ) ≡ 0

curl(curl A) ≡ grad(div A) − div(grad A) ≡ grad(div A) − ∇ 2 A grad( A1 · A2 ) ≡ A1 × (curl A2 ) + ( A1 · grad) A2 + A2 × (curl A1 ) + ( A2 · grad) A1

Page 5 of 24

Grad, Div, Curl and the Laplacian Cartesian Coordinates

Conversion to Cartesian Coordinates

Cylindrical Coordinates

x = ρ cos ϕ

Vector A

Axi + A y j + Az k

Gradient ∇φ

∂φ ∂φ ∂φ i+ j+ k ∂x ∂y ∂z

Divergence

∇·A

y = ρ sin ϕ

1 ∂φ ∂φ ∂φ b+ b+ b ρ ϕ z ∂ρ ρ ∂ϕ ∂z

i j k ∂ ∂ ∂ ∂x ∂y ∂z Ax A y Az

1 1 ρ b b b ϕ z ρ ρ ∂ ∂ ∂ ∂ρ ∂ϕ ∂z Aρ ρ Aϕ A z

∂ 2φ ∂2φ ∂2φ + 2 + 2 2 ∂x ∂y ∂z

∇2φ

b + Aϕϕ b Arbr + Aθθ

b + Aϕϕ b + Azb Aρ ρ z

1 ∂Aϕ ∂A z 1 ∂(ρ Aρ ) + + ρ ∂ρ ρ ∂ϕ ∂z

Laplacian

x = r cos ϕ sin θ y = r sin ϕ sin θ z = r cos θ

z=z

∂A y ∂A z ∂A x + + ∂x ∂y ∂z

Curl ∇ × A

Spherical Coordinates

1 ∂ ρ ∂ρ

1 ∂φ 1 ∂φ b ∂φ b br + θ+ ϕ ∂r r ∂θ r sin θ ∂ϕ

1 ∂Aθ sin θ 1 ∂ (r 2 Ar ) + 2 ∂r r sin θ ∂θ r 1 ∂Aϕ + r sin θ ∂ϕ 1 1 b 1 b 2 br θ ϕ r sin θ r sin θ r ∂ ∂ ∂ ∂r ∂θ ∂ϕ Ar rAθ rAϕ sin θ     1 1 ∂ ∂ ∂φ 2 ∂φ r + 2 sin θ ∂r ∂θ r2 ∂r r sin θ ∂θ

  ∂φ 1 ∂2φ ∂ 2φ ρ + 2 2+ 2 ∂ρ ρ ∂ϕ ∂z

+

Transformation of integrals L = the distance along some curve ‘C’ in space and is measured from some fixed point. S = a surface area

τ = a volume contained by a specified surface bt = the unit tangent to C at the point P b n = the unit outward pointing normal A = some vector function

dL = the vector element of curve (= bt dL)

dS = the vector element of surface (= b n dS) Z

Then

C

A · bt dL =

and when A = ∇φ Z

C

Z

Z

(∇φ) · dL =

C

C

A · dL

dφ

Gauss’s Theorem (Divergence Theorem) When S defines a closed region having a volume τ

also

Z

(∇ · A) dτ =

Z

(∇φ) dτ =

τ

τ

Z

Z

S

S

(A · b n) dS =

φ dS

Z

S

A · dS Z

τ

(∇ × A) dτ =

Page 6 of 24

Z

S

(b n × A) dS

∂ 2φ 1 2 r sin θ ∂ϕ2 2

Stokes’s Theorem When C is closed and bounds the open surface S, Z

also Z

S

S

(∇ × A) · dS =

Z

C

(b n × ∇φ) dS =

Z

C

A · dL

φ dL

Green’s Theorem Z

ψ∇φ · dS =

S

=

Z

∇ · (ψ∇φ) dτ

Zτ  τ

Green’s Second Theorem Z

τ

 ψ∇2φ + (∇ψ) · (∇φ) dτ

(ψ∇2φ − φ∇2 ψ) dτ =

Z

S

[ψ(∇φ) − φ(∇ψ)] · dS

Complex Variables Complex numbers The complex number z = x + iy = r(cos θ + i sin θ ) = r ei(θ +2nπ), where i2 = −1 and n is an arbitrary integer. The real quantity r is the modulus of z and the angle θ is the argument of z. The complex conjugate of z is z ∗ = x − iy = 2 r(cos θ − i sin θ ) = r e−iθ ; zz∗ = | z| = x2 + y2

De Moivre’s theorem (cos θ + i sin θ )n = einθ = cos nθ + i sin nθ

Power series for complex variables. ez

z3 3! z2 cos z =1− 2! z2 ln(1 + z) = z − 2

sin z

z2 zn + ···+ + ··· 2! n! z5 + −··· 5! z4 + − ··· 4! z3 + −··· 3

=1+z+ =z−

convergent for all finite z convergent for all finite z convergent for all finite z principal value of ln (1 + z)

This last series converges both on and within the circle | z| = 1 except at the point z = −1.

z3 z5 + −··· 3 5 This last series converges both on and within the circle | z| = 1 except at the points z = ±i. tan−1 z

=z−

n(n − 1) 2 n(n − 1)(n − 2) 3 z + z + ··· 2! 3! This last series converges both on and within the circle | z| = 1 except at the point z = −1.

(1 + z)n = 1 + nz +

Page 7 of 24

Trigonometric Formulae cos2 A + sin 2 A = 1

sec2 A − tan2 A = 1

cos 2A = cos 2 A − sin 2 A

sin 2A = 2 sin A cos A

cosec2 A − cot2 A = 1 2 tan A tan 2A = . 1 − tan2 A

sin ( A ± B) = sin A cos B ± cos A sin B

cos A cos B =

cos( A + B) + cos( A − B) 2

cos( A ± B) = cos A cos B ∓ sin A sin B

sin A sin B =

cos( A − B) − cos( A + B) 2

sin A cos B =

sin( A + B) + sin ( A − B) 2

tan( A ± B) =

tan A ± tan B 1 ∓ tan A tan B

sin A + sin B = 2 sin

A+B A−B cos 2 2

cos2 A =

1 + cos 2A 2

sin A − sin B = 2 cos

A+B A−B sin 2 2

sin 2 A =

1 − cos 2A 2

cos A + cos B = 2 cos

A+B A−B cos 2 2

cos3 A =

3 cos A + cos 3A 4

sin 3 A =

3 sin A − sin 3A 4

cos A − cos B = −2 sin

A−B A+B sin 2 2

Relations between sides and angles of any plane triangle In a plane triangle with angles A, B, and C and sides opposite a, b, and c respectively, a b c = = = diameter of circumscribed circle. sin A sin B sin C a2 = b2 + c2 − 2bc cos A a = b cos C + c cos B

b2 + c2 − a2 2bc a−b C A−B = cot tan 2 a+b 2 q 1 1 1 area = ab sin C = bc sin A = ca sin B = s(s − a)(s − b)(s − c), 2 2 2 cos A =

where s =

Relations between sides and angles of any spherical triangle In a spherical triangle with angles A, B, and C and sides opposite a, b, and c respectively, sin a sin b sin c = = sin A sin B sin C cos a = cos b cos c + sin b sin c cos A cos A = − cos B cos C + sin B sin C cos a

Page 8 of 24

1 ( a + b + c) 2

Hyperbolic Functions x2 x4 1 x ( e + e− x ) = 1 + + + ··· 2 2! 4! 1 x3 x5 sinh x = ( ex − e− x ) = x + + + ··· 2 3! 5!

valid for all x

cosh x =

valid for all x

cosh ix = cos x

cos ix = cosh x

sinh ix = i sin x sinh x tanh x = cosh x cosh x coth x = sinh x

sin ix = i sinh x 1 sech x = cosh x 1 cosech x = sinh x

cosh 2 x − sinh 2 x = 1

For large positive x: cosh x ≈ sinh x →

ex 2

tanh x → 1

For large negative x: cosh x ≈ − sinh x →

e− x 2

tanh x → −1

Relations of the functions sinh x

= − sinh (− x)

sech x

cosh x

= cosh (− x)

cosech x = − cosech(− x)

tanh x

= − tanh(− x)

coth x

sinh x =

tanh x

2 tanh ( x/2) 2

1 − tanh ( x/2)

=

q

q

2

=q

tanh x 1 − tanh2 x

1 − sech x

cosh x =

sech x

= sech(− x)

= − coth (− x) 1 + tanh2 ( x/2) 2

1 − tanh ( x/2)

=

cosech 2 x + 1 r cosh x − 1 sinh ( x/2) = 2 cosh x − 1 sinh x tanh( x/2) = = sinh x cosh x + 1

cosech x =

sinh (2x) = 2 sinh x cosh x

tanh(2x) =

coth x

=

q

q

1 − tanh2 x

= q

coth 2 x − 1 r cosh x + 1 cosh ( x/2) = 2

2 tanh x 1 + tanh 2 x

cosh (2x) = cosh 2 x + sinh 2 x = 2 cosh2 x − 1 = 1 + 2 sinh 2 x sinh (3x) = 3 sinh x + 4 sinh 3 x tanh(3x) =

3 tanh x + tanh3 x

cosh 3x = 4 cosh 3 x − 3 cosh x

1 + 3 tanh2 x

Page 9 of 24

1 1 − tanh2 x

sinh ( x ± y) = sinh x cosh y ± cosh x sinh y cosh( x ± y) = cosh x cosh y ± sinh x sinh y tanh( x ± y) =

tanh x ± tanh y 1 ± tanh x tanh y

1 sinh x + sinh y = 2 sinh ( x + y) cosh 2 1 sinh x − sinh y = 2 cosh ( x + y) sinh 2 sinh x ± cosh x = tanh x ± tanh y =

1 ( x − y) 2 1 ( x − y) 2

1 ± tanh ( x/2) = e± x 1 ∓ tanh( x/2) sinh ( x ± y) cosh x cosh y

coth x ± coth y = ±

Inverse functions

sinh ( x ± y) sinh x sinh y

! x2 + a2 sinh a ! p x + x2 − a2 −1 x cosh = ln a a   1 a+x −1 x tanh = ln a 2 a−x   x 1 x +a −1 coth = ln a 2 x−a   s 2 x a a sech−1 = ln  + − 1 a x x2   s 2 a a x + 1 cosech−1 = ln  + a x x2 −1

x = ln a

1 1 cosh x + cosh y = 2 cosh ( x + y) cosh ( x − y) 2 2 1 1 cosh x − cosh y = 2 sinh ( x + y) sinh ( x − y) 2 2

x+

p

for −∞ < x < ∞ for x ≥ a for x2 < a2 for x2 > a2 for 0 < x ≤ a for x 6= 0

Limits nc xn → 0 as n → ∞ if | x| < 1 (any fixed c) xn /n! → 0 as n → ∞ (any fixed x)

(1 + x/n)n → ex as n → ∞, x ln x → 0 as x → 0 If f ( a) = g( a) = 0

12

then

lim

x→ a

f 0 ( a) f ( x) = 0 g( x) g ( a)

(l’Hopital’s ˆ rule)

Page 10 of 24

Differentiation (uv)0 = u0 v + uv0 ,

 u 0 v

=

u0 v − uv0 v2

(uv)(n) = u(n) v + nu(n−1) v(1) + · · · + n Cr u(n−r) v(r) + · · · + uv(n)   n! n n = where Cr ≡ r!(n − r)! r d (sin x) = cos x dx d (cos x) = − sin x dx d (tan x) = sec2 x dx d (sec x) = sec x tan x dx d (cot x) = − cosec2 x dx d (cosec x) = − cosec x cot x dx

d (sinh x) dx d (cosh x) dx d (tanh x) dx d (sech x) dx d (coth x) dx d (cosech x) dx

Leibniz Theorem

= cosh x = sinh x = sech2 x = − sech x tanh x = − cosech2 x = − cosech x coth x

Integration Standard forms Z

xn dx =

1 dx x Z eax dx Z

Z Z Z Z Z Z Z Z Z Z

xn+1 +c n+1

= ln x + c

for n 6= −1 Z

ln x dx = x(ln x − 1) + c   1 x ax ax x e dx = e − 2 +c a a

1 ax e +c a   x2 1 x ln x dx = ln x − +c 2 2 x 1 1 dx = tan−1 +c 2 2 a a a +x     a+x 1 1 1 −1 x tanh ln dx = + c = +c a a 2a a−x a2 − x2     1 1 1 x−a −1 x dx = − coth +c= +c ln a a 2a x+a x2 − a2 x −1 1 dx = +c 2(n − 1) ( x2 ± a2 )n−1 ( x2 ± a2 )n x 1 dx = ln( x2 ± a2 ) + c 2 x2 ± a2 x 1 p dx = sin−1 +c a a2 − x2   p 1 p dx = ln x + x2 ± a2 + c x2 ± a2 p x p dx = x2 ± a2 + c x2 ± a2  x i p 1h p 2 a2 − x2 dx = x a − x2 + a2 sin −1 +c 2 a

=

Z

Page 11 of 24

for x2 < a2 for x2 > a2 for n 6= 1

Z Z

∞ 0

1 dx = π cosec pπ (1 + x) x p

∞

2

cos( x ) dx =

0

Z

∞ 0

for p < 1

1 sin ( x ) dx = 2 2

r

π 2

√ exp(− x2 /2σ 2 ) dx = σ 2π −∞  √ Z ∞  1 × 3 × 5 × · · · (n − 1)σ n+1 2π n 2 2 x exp(− x /2σ ) dx =  −∞ 0

Z

Z

Z Z Z Z Z

∞

sin x dx

cos x dx tan x dx

cot x dx

for n ≥ 1 and odd

Z

sinh x dx

= cosh x + c

= sin x + c

Z

cosh x dx

= sinh x + c

= − ln(cos x) + c

Z

tanh x dx

= ln(cosh x) + c

Z

cosech x dx = ln [tanh( x/2)] + c

= ln(sec x + tan x) + c

Z

sech x dx

= 2 tan−1 ( ex ) + c

= ln(sin x) + c

Z

coth x dx

= ln(sinh x) + c

= − cos x + c

cosec x dx = ln(cosec x − cot x) + c sec x dx

for n ≥ 2 and even

sin (m + n) x sin (m − n) x − +c 2(m − n) 2(m + n) Z sin (m + n) x sin (m − n) x + +c cos mx cos nx dx = 2(m − n) 2(m + n) Z

sin mx sin nx dx =

if m2 6= n2 if m2 6= n2

Standard substitutions If the integrand is a function of: p ( a2 − x2 ) or a2 − x2 p ( x2 + a2 ) or x2 + a2 p ( x2 − a2 ) or x2 − a2

substitute: x = a sin θ or x = a cos θ x = a tan θ or x = a sinh θ x = a sec θ or x = a cosh θ

If the integrand is a rational function of sin x or cos x or both, substitute t = tan( x/2) and use the results: sin x =

2t 1 + t2

cos x =

1 − t2 1 + t2

If the integrand is of the form: Z Z

dx p ( ax + b) px + q

dx q ( ax + b) px2 + qx + r

dx =

2 dt . 1 + t2

substitute: px + q = u2

ax + b =

1 . u

Page 12 of 24

Integration by parts Z

b a

b Z b u dv = uv − v du a a

Differentiation of an integral If f ( x, α ) is a function of x containing a parameter α and the limits of integration a and b are functions of α then Z b(α )

d dα

a (α )

f ( x, α ) dx = f (b, α )

db da − f ( a, α ) + dα dα

Z b(α ) ∂

f ( x, α ) dx.

∂α

a (α )

Special case, d dx

Z

x a

f ( y) dy = f ( x).

Dirac δ-‘function’ 1 δ (t − τ ) = 2π

Z

∞

−∞

exp[iω(t − τ )] dω.

If f (t) is an arbitrary function of t then

δ (t) = 0 if t 6= 0, also

Z

∞

−∞

Z

∞

−∞

δ (t − τ ) f (t) dt = f (τ ).

δ (t) dt = 1

Reduction formulae Factorials n! = n(n − 1)(n − 2) . . . 1,

0! = 1.

Stirling’s formula for large n: For any p > −1,

Z

∞ 0

For any p, q > −1,

Z

x p e− x dx = p 1

0

ln(n!) ≈ n ln n − n. Z

∞

0

x p (1 − x)q dx =

x p−1 e− x dx = p!.

(− 1/2)! =

√

π,

( 1/2)! =

√ π/ , 2

p!q! . ( p + q + 1)!

Trigonometrical If m, n are integers, m − 1 π/ 2 n − 1 π/ 2 sin m−2 θ cosn θ dθ = sin m θ cosn−2 θ dθ m+n 0 m+n 0 0 and can therefore be reduced eventually to one of the following integrals Z π/ 2

sin m θ cos n θ dθ =

Z π/ 2

sin θ cos θ dθ =

0

1 , 2

Z

Z

Z π/ 2 0

sin θ dθ = 1,

Z π/ 2 0

cos θ dθ = 1,

Z π/ 2 0

Other If In =

Z

∞ 0

xn exp(−α x2 ) dx

then

In =

(n − 1) In − 2 , 2α

I0 =

Page 13 of 24

1 2

r

π , α

I1 =

1 . 2α

dθ =

π . 2

etc.

Differential Equations Diffusion (conduction) equation ∂ψ = κ ∇2ψ ∂t

Wave equation ∇2ψ =

1 ∂2ψ c2 ∂t2

Legendre’s equation (1 − x2 )

dy d2 y − 2x + l (l + 1) y = 0, dx2 dx

1 solutions of which are Legendre polynomials Pl ( x), where Pl ( x) = l 2 l! 1 2 P0 ( x) = 1, P1 ( x) = x, P2 ( x) = (3x − 1) etc. 2 Recursion relation Pl ( x) =

1 [(2l − 1) xPl −1 ( x) − (l − 1) Pl −2( x)] l

Orthogonality Z

1

−1

Pl ( x) Pl 0 ( x) dx =

2 δll 0 2l + 1

Bessel’s equation x2

d2 y dy +x + ( x2 − m2 ) y = 0, dx2 dx

solutions of which are Bessel functions Jm ( x) of order m. Series form of Bessel functions of the first kind

(−1)k ( x/2)m+2k k!(m + k)! k=0 ∞

Jm ( x ) =

∑

(integer m).

The same general form holds for non-integer m > 0.

Page 14 of 24



d dx

l


l x2 − 1 , Rodrigues’ formula so

Laplace’s equation ∇2 u = 0

If expressed in two-dimensional polar coordinates (see section 4), a solution is    u(ρ, ϕ) = Aρn + Bρ−n C exp(inϕ) + D exp(−inϕ)

where A, B, C, D are constants and n is a real integer.

If expressed in three-dimensional polar coordinates (see section 4) a solution is     u(r, θ , ϕ) = Arl + Br−(l +1) Plm C sin mϕ + D cos mϕ where l and m are integers with l ≥ |m| ≥ 0; A, B, C, D are constants;  | m | d Plm (cos θ ) = sin|m| θ Pl (cos θ ) d(cos θ )

is the associated Legendre polynomial. Pl0 (1) = 1.

If expressed in cylindrical polar coordinates (see section 4), a solution is    u(ρ, ϕ, z) = Jm (nρ) A cos mϕ + B sin mϕ C exp(nz) + D exp(−nz)

where m and n are integers; A, B, C, D are constants.

Spherical harmonics The normalized solutions Ylm (θ , ϕ) of the equation     1 ∂ ∂ ∂2 1 sin θ + Ylm + l (l + 1)Ylm = 0 sin θ ∂θ ∂θ sin2 θ ∂ϕ2 are called spherical harmonics, and have values given by s  m 2l + 1 (l − |m|)! m for m ≥ 0 m imϕ Yl (θ , ϕ) = × (−1) Pl (cos θ ) e 4π (l + |m|)! 1 for m < 0 r r r 1 3 3 0 ±1 0 i.e., Y0 = , Y1 = cos θ, Y1 = ∓ sin θ e±iϕ , etc. 4π 4π 8π Orthogonality Z

Yl∗m Ylm0 dΩ = δll 0 δmm0 0

4π

Calculus of Variations The condition for I =

Z

b a

F ( y, y0, x) dx to have a stationary value is

Euler–Lagrange equation.

Page 15 of 24

∂F d = ∂y dx



 dy ∂F 0 . This is the 0 , where y = dx ∂y

Functions of Several Variables ∂φ implies differentiation with respect to x keeping y, z, . . . constant. ∂x ∂φ ∂φ ∂φ ∂φ ∂φ ∂φ dφ = dx + dy + dz + · · · and δφ ≈ δx + δy + δz + · · · ∂x ∂y ∂z ∂x ∂y ∂z   ∂φ ∂φ ∂φ is also written as when the variables kept where x, y, z, . . . are independent variables. or ∂x ∂x ∂x

If φ = f ( x, y, z, . . .) then

y,...

constant need to be stated explicitly. If φ is a well-behaved function then

If φ = f ( x, y),   ∂φ 1 =   , ∂x ∂x y ∂φ y



∂φ ∂x

y,...

∂2φ ∂ 2φ = etc. ∂x ∂y ∂y ∂x

  y

∂x ∂y

  φ

∂y ∂φ



x

= −1.

Taylor series for two variables If φ( x, y) is well-behaved in the vicinity of x = a, y = b then it has a Taylor series  2 2  ∂φ ∂2φ ∂φ 1 2∂ φ 2∂ φ +··· φ( x, y) = φ( a + u, b + v) = φ( a, b) + u u + 2uv +v + +v ∂x ∂y 2! ∂x ∂y ∂x2 ∂y2

where x = a + u, y = b + v and the differential coefficients are evaluated at x = a,

y=b

Stationary points ∂2φ ∂2φ ∂φ ∂φ ∂2φ = = = 0. Unless 2 = = 0, the following 2 ∂x ∂y ∂x ∂y ∂x ∂y conditions determine whether it is a minimum, a maximum or a saddle point.  ∂2φ ∂2φ   > 0, or > 0,   2 2 Minimum:  ∂2φ ∂2φ ∂φ ∂x2 ∂y2 and > 2 2 2 2  ∂x ∂y ∂φ ∂φ ∂x ∂y  Maximum: < 0, or < 0,   2 2 ∂x ∂y  2 ∂2φ ∂2φ ∂2φ Saddle point: < ∂x ∂y ∂x2 ∂y2

A function φ = f ( x, y) has a stationary point when

If

∂2φ ∂2φ ∂2φ = = = 0 the character of the turning point is determined by the next higher derivative. ∂x ∂y ∂x2 ∂y2

Changing variables: the chain rule If φ = f ( x, y, . . .) and the variables x, y, . . . are functions of independent variables u, v, . . . then ∂φ ∂φ ∂x ∂φ ∂y = + + ··· ∂u ∂x ∂u ∂y ∂u ∂φ ∂x ∂φ ∂y ∂φ = + + ··· ∂v ∂x ∂v ∂y ∂v etc.

Page 16 of 24

Changing variables in surface and volume integrals – Jacobians If an area A in the x, y plane maps into an area A 0 in the u, v plane then ∂x ∂x Z Z ∂u ∂v f ( x, y) dx dy = f (u, v) J du dv where J = A A0 ∂y ∂y ∂u ∂v ∂( x, y) The Jacobian J is also written as . The corresponding formula for volume integrals is ∂(u, v) ∂x ∂x ∂x ∂u ∂v ∂w Z Z ∂y ∂y ∂y f ( x, y, z) dx dy dz = f (u, v, w) J du dv dw where now J = V V0 ∂u ∂v ∂w ∂z ∂z ∂z ∂u ∂v ∂w

Fourier Series and Transforms Fourier series If y( x) is a function defined in the range −π ≤ x ≤ π then y( x) ≈ c0 +

M

∑

cm cos mx +

m=1

M0

∑

sm sin mx

m=1

where the coefficients are Z π 1 y( x) dx c0 = 2π −π Z π 1 cm = y( x) cos mx dx π −π Z π 1 sm = y( x) sin mx dx π −π

(m = 1, . . . , M) (m = 1, . . . , M 0 )

with convergence to y( x) as M, M 0 → ∞ for all points where y( x) is continuous.

Fourier series for other ranges Variable t, range 0 ≤ t ≤ T, (i.e., a periodic function of time with period T, frequency ω = 2π/ T). y(t) ≈ c0 + ∑ cm cos mωt + ∑ sm sin mωt

where

ω T ω T ω T y(t) dt, cm = y(t) cos mωt dt, sm = y(t) sin mωt dt. 2π 0 π 0 π 0 Variable x, range 0 ≤ x ≤ L, 2mπx 2mπx y( x) ≈ c0 + ∑ cm cos + ∑ sm sin L L where Z Z Z 2 L 1 L 2 L 2mπx 2mπx dx, sm = dx. c0 = y( x) dx, cm = y( x) cos y( x) sin L 0 L 0 L L 0 L c0 =

Z

Z

Z

Page 17 of 24

Fourier series for odd and even functions If y( x) is an odd (anti-symmetric) function [i.e., y(− x) = − y( x)] defined in the range −π ≤ x ≤ π, then only Z 2 π sines are required in the Fourier series and s m = y( x) sin mx dx. If, in addition, y( x) is symmetric about π 0 Z 4 π/ 2 y( x) sin mx dx (for m odd). If x = π/2, then the coefficients s m are given by sm = 0 (for m even), s m = π 0 y( x) is an even (symmetric) function [i.e., y(− x) = y( x)] defined in the range −π ≤ x ≤ π, then only constant Z Z 2 π 1 π y( x) dx, cm = y( x) cos mx dx. If, in and cosine terms are required in the Fourier series and c 0 = π 0 π 0 π addition, y( x) is anti-symmetric about x = , then c0 = 0 and the coefficients c m are given by cm = 0 (for m even), 2 Z 4 π/ 2 cm = y( x) cos mx dx (for m odd). π 0 [These results also apply to Fourier series with more general ranges provided appropriate changes are made to the limits of integration.]

Complex form of Fourier series If y( x) is a function defined in the range −π ≤ x ≤ π then M

∑

y( x) ≈

−M

Cm eimx ,

Cm =

1 2π

Z

π

y( x) e−imx dx

−π

with m taking all integer values in the range ± M. This approximation converges to y( x) as M → ∞ under the same conditions as the real form. For other ranges the formulae are: Variable t, range 0 ≤ t ≤ T, frequency ω = 2π/ T, ∞

y(t) =

∑ Cm e

imω t

−∞

,

ω Cm = 2π

Variable x0 , range 0 ≤ x0 ≤ L, ∞

0

y( x ) =

∑ Cm e

i2mπx0 / L

,

−∞

Z

T

0

1 Cm = L

y(t) e−imωt dt.

Z

L 0

y( x0 ) e−i2mπx / L dx0 . 0

Discrete Fourier series If y( x) is a function defined in the range −π ≤ x ≤ π which is sampled in the 2N equally spaced points x n = nx/ N [n = −( N − 1) . . . N ], then y( xn ) = c0 + c1 cos xn + c2 cos 2xn + · · · + c N −1 cos( N − 1) xn + c N cos Nxn

+ s1 sin xn + s2 sin 2xn + · · · + s N −1 sin ( N − 1) xn + s N sin Nxn

where the coefficients are 1 y( xn ) c0 = 2N ∑ 1 cm = y( xn ) cos mxn N∑ 1 cN = y( xn ) cos Nxn 2N ∑ 1 sm = y( xn ) sin mxn N∑ 1 y( xn ) sin Nxn sN = 2N ∑ each summation being over the 2N sampling points x n .

Page 18 of 24

(m = 1, . . . , N − 1)

(m = 1, . . . , N − 1)

Fourier transforms If y( x) is a function defined in the range −∞ ≤ x ≤ ∞ then the Fourier transform b y(ω) is defined by the equations Z ∞ Z ∞ 1 b by(ω) = y(ω) eiωt dω, y(t) e−iωt dt. y(t) = 2π −∞ −∞ If ω is replaced by 2π f , where f is the frequency, this relationship becomes y(t) =

Z

∞

−∞

by( f ) ei2π f t d f ,

by( f ) =

Z

∞

−∞

y(t) e−i2π f t dt.

If y(t) is symmetric about t = 0 then Z Z ∞ 1 ∞ by(ω) cos ωt dω, by(ω) = 2 y(t) = y(t) cos ωt dt. π 0 0 If y(t) is anti-symmetric about t = 0 then Z Z ∞ 1 ∞ by(ω) sin ωt dω, by(ω) = 2 y(t) sin ωt dt. y(t) = π 0 0

Specific cases

y(t) = a, = 0,

|t| ≤ τ |t| > τ



y(t) = a(1 − |t|/τ ), = 0,

y(t) = exp(−t2 /t20 )

y(t) = f (t) eiω0 t

by(ω) = 2a

(‘Top Hat’),

|t| ≤ τ |t| > τ



(‘Saw-tooth’),

∑

m =− ∞

where

(modulated function),

by(ω) = bf (ω − ω0 ) ∞

δ (t − mτ ) (sampling function)

by(ω) =

Page 19 of 24

sinc( x) =

  2a 2 ωτ ( 1 − cos ωτ ) = a τ sinc 2 ω2 τ

√
by(ω) = t0 π exp −ω2 t20 /4

(Gaussian),

∞

y(t) =

by(ω) =

sin ωτ ≡ 2aτ sinc (ωτ ) ω

∑

n =− ∞

δ (ω − 2πn/τ )

sin ( x) x

Convolution theorem If z(t) =

Z

∞

−∞

x(τ ) y(t − τ ) dτ =

Z

∞

−∞

x(t − τ ) y(τ ) dτ

≡ x(t) ∗ y(t) then

Conversely, xcy = b x ∗ by.

bz (ω) = b x(ω) by(ω).

Parseval’s theorem Z

∞

−∞

y∗ (t) y(t) dt =

1 2π

Z

∞

−∞

by∗ (ω) by(ω) dω

(if b y is normalised as on page 21)

Fourier transforms in two dimensions b (k) = V

=

Z

V (r ) e−ik·r d2 r

Z

∞ 0

2πrV (r) J0 (kr) dr

if azimuthally symmetric Examples

Fourier transforms in three dimensions b (k) = V

Z

V (r )

V (r ) e−ik·r d3 r

4π ∞ = V (r) r sin kr dr k 0 Z 1 b (k) eik·r d3 k V (r ) = V (2π)3 Z

if spherically symmetric

1 4πr e− λ r 4πr ∇V (r )

∇ 2 V (r )

Page 20 of 24

b (k) V

1 k2 1 2 k + λ2 b (k) ikV b (k) −k2 V

Laplace Transforms If y(t) is a function defined for t ≥ 0, the Laplace transform y(s) is defined by the equation y(s) = L{ y(t)} =

Z

∞

0

e−st y(t) dt

Function y(t) (t > 0)

Transform y(s)

δ (t)

1

Delta function

θ (t)

1 s

Unit step function

n! sn+1 r 1 π 2 s3 r π s

tn 1

t /2 1

t− /2

1 (s + a)

e− at sin ωt

(s2

s (s2 + ω2 ) ω (s2 − ω2 ) s (s2 − ω2 )

cos ωt sinh ωt cosh ωt e− at y(t)

y( s + a )

e − sτ y ( s )

y(t − τ ) θ (t − τ ) ty(t)

−

dy dt dn y dtn Z Z Z

t 0 t 0

t 0

y(τ ) dτ

   x(τ ) y(t − τ ) dτ      x(t − τ ) y(τ ) dτ 

ω + ω2

dy ds

s y( s ) − y ( 0 ) n

s y( s ) − s

n−1

y(0) − s

n−2



dy dt





dn−1 y ···− dtn−1 0



0

y( s ) s

x ( s ) y( s )

[Note that if y(t) = 0 for t < 0 then the Fourier transform of y(t) is by(ω) = y(iω).]

Page 21 of 24

Convolution theorem

Numerical Analysis Finding the zeros of equations If the equation is y = f ( x) and x n is an approximation to the root then either f ( xn ) . xn+1 = xn − 0 f ( xn ) xn − xn−1 or, xn+1 = xn − f ( xn ) f ( xn ) − f ( xn−1 )

(Newton) (Linear interpolation)

are, in general, better approximations.

Numerical integration of differential equations If

dy = f ( x, y) then dx yn+1 = yn + h f ( xn , yn ) where h = xn+1 − xn

(Euler method)

y∗n+1 = yn + h f ( xn , yn ) h[ f ( xn , yn ) + f ( xn+1 , y∗n+1 )] yn+1 = yn + 2

Putting then

(improved Euler method)

Central difference notation If y( x) is tabulated at equal intervals of x, where h is the interval, then δy n+1/2 = yn+1 − yn and δ2 yn = δyn+1/2 − δyn−1/2

Approximating to derivatives 



dy dx



d2 y dx2

n



≈ n

≈

δy 1 + δyn− 1/2 yn+1 − yn yn − yn−1 ≈ ≈ n+ /2 h h 2h

where h = xn+1 − xn

δ2 y n yn+1 − 2yn + yn−1 = h2 h2

Interpolation: Everett’s formula y( x) = y( x0 + θ h) ≈ θ y0 + θ y1 +

1 1 2 θ (θ − 1)δ2 y0 + θ (θ 2 − 1)δ2 y1 + · · · 3! 3!

where θ is the fraction of the interval h (= x n+1 − xn ) between the sampling points and θ = 1 − θ. The first two terms represent linear interpolation.

Numerical evaluation of definite integrals Trapezoidal rule The interval of integration is divided into n equal sub-intervals, each of width h; then   Z b 1 1 f ( x) dx ≈ h c f ( a) + f ( x1 ) + · · · + f ( x j ) + · · · + f (b) 2 2 a where h = (b − a)/n and x j = a + jh. Simpson’s rule The interval of integration is divided into an even number (say 2n) of equal sub-intervals, each of width h = (b − a)/2n; then Z b  h f ( a) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + · · · + 2 f ( x2n−2 ) + 4 f ( x2n−1 ) + f (b) f ( x) dx ≈ 3 a

Page 22 of 24

Gauss’s integration formulae These have the general form For n = 2 : For n = 3 :

xi = ±0·5773;

Z

1

−1

n

y( x) dx ≈ ∑ ci y( xi ) 1

c i = 1, 1 (exact for any cubic).

xi = −0·7746, 0·0, 0·7746;

c i = 0·555, 0·888, 0·555 (exact for any quintic).

Treatment of Random Errors Sample mean

x=

1 ( x1 + x2 + · · · xn ) n

Residual:

d=x−x 1 Standard deviation of sample: s = √ (d21 + d22 + · · · d2n )1/2 n 1 Standard deviation of distribution: σ ≈ √ (d21 + d22 + · · · d2n )1/2 n−1 σ 1 σm = √ = q (d21 + d22 + · · · d2n )1/2 Standard deviation of mean: n n ( n − 1)

Result of n measurements is quoted as x ± σ m .

=q

1

n ( n − 1)



∑

x2i

1 − n

∑ xi


2

1 / 2

Range method A quick but crude method of estimating σ is to find the range r of a set of n readings, i.e., the difference between the largest and smallest values, then r σ≈ √ . n This is usually adequate for n less than about 12.

Combination of errors If Z = Z ( A, B, . . .) (with A, B, etc. independent) then  2  2 ∂Z ∂Z σA + σB + · · · (σ Z )2 = ∂A ∂B

So if (i)

Z = A ± B ± C,

(ii)

Z = AB or A/ B,

(iii)

Z = Am ,

(iv)

Z = ln A,

(v)

Z = exp A,

(σ Z )2 = (σ A )2 + (σ B )2 + (σC )2  σ 2  σ  2  σ  2 Z B A = + Z A B σZ σ =m A Z A σA σZ = A σZ = σA Z

Page 23 of 24

Statistics Mean and Variance A random variable X has a distribution over some subset x of the real numbers. When the distribution of X is discrete, the probability that X = x i is Pi . When the distribution is continuous, the probability that X lies in an interval δx is f ( x)δx, where f ( x) is the probability density function. Mean µ = E( X ) = ∑ Pi xi or

Z

x f ( x) dx.

Variance σ 2 = V ( X ) = E[( X − µ )2 ] =

∑ Pi (xi − µ )2 or

Z

( x − µ )2 f ( x) dx.

Probability distributions Error function: Binomial: Poisson: Normal:

x 2 2 e− y dy erf( x) = √ π 0   n x n− x f ( x) = p q where q = (1 − p), x

Z

µ = np, σ 2 = npq, p < 1.

µ x −µ e , and σ 2 = µ x!   1 ( x − µ )2 f ( x) = √ exp − 2σ 2 σ 2π f ( x) =

Weighted sums of random variables If W = aX + bY then E(W ) = aE( X ) + bE(Y ). If X and Y are independent then V (W ) = a 2 V ( X ) + b2 V (Y ).

Statistics of a data sample x 1 , . . . , xn 1 n

Sample mean x =

∑ xi

1 Sample variance s = n 2

∑( x i − x )

2

=



1 x2 n∑ i



− x2 = E( x2 ) − [E( x)]2

Regression (least squares fitting) To fit a straight line by least squares to n pairs of points ( x i , yi ), model the observations by y i = α + β( xi − x) + i , where the i are independent samples of a random variable with zero mean and variance σ 2 . Sample statistics: s 2x =

1 n

∑( x i − x ) 2 ,

s2xy

s2y =

1 n

∑ ( y i − y) 2 ,

s2xy =

1 n

∑(xi − x)( yi − y).

n (residual variance), n−2 s4 1 b ( xi − x)}2 = s2 − xy . b −β where residual variance = ∑{ yi − α y n s2x

b= b = y, β Estimators: α

s2x

b ( x − x); σb 2 = b+β ; E(Y at x) = α

b2 σb 2 b are σ b and β Estimates for the variances of α and 2 . n ns x b=r= Correlation coefficient: ρ

s2xy

sx s y

.

Page 24 of 24