Mathematical Formula Booklet for Physics

RHUL

1

Contents

1 Introduction 6

1.1 Useful references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Astrophysical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Series 8

2.1 Arithmetic and geometric progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Convergence of series: the ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Power series expansions with real variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Integer series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Plane wave expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Vector Identities 10

3.1 Scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Equation of a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Equation of a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.5 Scalar triplet product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.6 Vector triplet product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.7 Non-orthogonal basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.8 Summation convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Matrix Algebra 11

4.1 Unit matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 Transpose matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2

4.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.5 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.6 2× 2 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7 Product rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.8 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.9 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Vector Calculus 13

5.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.3 Grad, Div, Curl and the Laplacian in Cartesian, cylindrical and spherical coordinates. . . . . . . . . . . . 15

5.4 Transformation of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.4.1 Gauss’s theorem (divergence theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.4.2 Stoke’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.4.3 Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.4.4 Green’s second theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Complex Variables 16

6.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.2 De Moivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.2.1 Complex logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.3 Cauchy’s residue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.4 Power series of complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.4.1 Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Trigonometric Formulae 18

7.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.2 Relations between sides and angles of any plane triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.3 Relations between sides and angles of any spherical triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Hyperbolic Functions 20

3

8.1 Relations of the functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.2 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Limits 22

10 Differentiation 22

11 Integration 23

11.1 Standard forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11.2 Standard substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

11.3 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.4 Differentiation of an Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.5 Dirac δ− function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11.6 Reduction formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.6.1 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.6.2 Trigonometrical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11.6.3 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12 Differential Equations 27

12.1 Diffusion (conduction) equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12.3 Legendre’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12.4 Bessel’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

12.5 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

12.6 Spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

13 Calculus of Variations 29

14 Functions of Several Variables 29

14.1 Taylor series for two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

14.2 Changing variables: the chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

14.3 Changing variables in integrals - Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4

15 Fourier Series and Transforms 30

15.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

15.2 Fourier series for other ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.3 Fourier series for odd and even functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.4 Complex form of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.5 Discrete Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15.6 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15.6.1 Specific cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15.7 Convolution theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15.8 Parseval’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15.9 Fourier transforms in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15.10Fourier transforms in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

16 Numerical Analysis 34

16.1 Finding the zeros of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16.2 Numerical integration of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16.3 Central difference notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16.4 Approximating to derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16.5 Numerical evaluation of definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

16.5.1 Trapezoidal rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

16.5.2 Simpson’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

16.5.3 Gauss’s integration formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

17 Statistics and Treatment of Random Errors 35

17.1 Mean, variance and covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

17.2 Error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

17.3 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

17.4 Method of least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5

1 Introduction

This mathematical formula handbook has been prepared in response to discussions in the curriculum development group

and the Undergraduate Committee, with the hope that it will be useful to those studying physics.

Please send suggestions for amendments to the Stephen.West@rhul.ac.uk, and they will be considered for incorporation

in the next edition. Please also let us know if any errors are spotted.

Version 1.0 September 2017.

1.1 Useful references

Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover, 1965.

Gradshteyn, I.S. and Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, 1980.

Jahnke, E. and Emde, F., Tables of Functions, Dover, 1986.

Nordling, C. and Osterman, J., Physics Handbook, Chartwell-Bratt, Bromley, 1980.

Spiegel, M.R., Mathematical Handbook of Formulas and Tables. (Schaum’s Outline Series, McGraw-Hill, 1968).

1.2 Abbreviations

A ampere Pa pascal

C coulomb s second

eV electron volt T tesla

F farad V volt

g gramme W watt

H henry Wb weber

Hz hertz mol gramme mole

J joule K kelvin

m metre Ω ohm

N newton S siemens (ohm−1)

1.3 Prefixes

T = tera = 1012 c = centi = 10−2

G = giga = 109 m = milli = 10−3

M = mega = 106 µ = micro = 10−6

k = kilo = 103 n = nano = 10−9

h = hecto = 102 p = pico = 10−12

f = femto = 10−15

6

1.4 Physical constants

Speed of light in vacuum c 2.997 924 58× 108 m s−1

Permeability of vacuum µ0 4π × 10−7 H m−1

Permittivity of vacuum ε0 1/µ0c2 = 8.854 187 817 . . .× 10−12 F m−1

Elementary charge e 1.602 176 620 8(98)× 10−19 C

Electron (rest) mass me 9.109 383 56(11)× 10−31 kg

Unified atomic mass constant mu 1.660 539 040(20)× 10−27 kg

Proton rest mass mp 1.672 621 898(21)× 10−27 kg

Planck constant h 6.626 070 040(81)× 10−34 J s

h/2π ~ 1.054 571 800(13)× 10−34 J s

~c 197.33 MeV fm

Boltzmann constant k 1.380 648 52(79)× 10−23 J K−1

Stefan-Boltzmann constant σ 5.670 367(13)× 10−8 W m−2 K−4

Molar gas constant R 8.314 459 8(48) J mol−1 K−1

Avogadro constant NA 6.022 140 857(74)× 1023 mol −1

Gravitational constant G 6.674 08(31)× 10−11 N m2 kg−2

Acceleration of free fall g 9.806 65 m s−2 (standard value at sea level)

Volume of one mole of an ideal gas at STP 2.241 4× 10−2 m3

One standard atmosphere P0 1.013 25× 105 N m−2

Bohr magneton eh/4πme µB 9.274 009 994(57)× 10−24 J T−1

Bohr Radius a0 5.292× 10−11 m

Fine Structure Constant e2/(4πε0~c) = α (137.04)−1

Compton Wavelength of electron h/(mec) = λC 2.4263× 10−12 m

Rydberg’s constant R∞ 1.0974× 107 m−1

R∞hc 13.606 eV

1.5 Astrophysical data

1 astronomical unit AU 1.496× 1011 m

1 parsec pc 3.086× 1016 m

Luminosity of Sun L 3.845× 1026 W

Mass of Sun M 1.988× 1030 kg

Radius of Sun R 6.955× 108 m

7

2 Series

2.1 Arithmetic and geometric progression

A.P. Sn = a+ (a+ d) + (a+ 2d) + . . .+ [a+ (n− 1)d] =n

2[2a+ (n− 1)d] ,

G.P. Sn = a+ ar + ar2 + . . .+ arn−1 = a1− rn

1− r,

(S∞ =

a

1− rfor |r| < 1.

)

These results also hold for complex series.

2.1.1 Convergence of series: the ratio test

Sn = u1 + u2 + u2 + . . .+ un converges as n→∞ if limn→∞

∣∣∣∣un+1

un

∣∣∣∣ < 1.

2.2 Taylor series

If y(x) is well-behaved in the vicinity of x = a then it has a Taylor series,

y(x) = y(a+ u) = y(a) + udy

dx

∣∣∣∣x=a

+u2

2!

d2y

dx2

∣∣∣∣x=a

+u3

3!

d3y

dx3

∣∣∣∣x=a

+ . . . ,

where u = x− a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with a = 0,

y(x) = y(0) + xdy

dx

∣∣∣∣x=0

+x2

2!

d2y

dx2

∣∣∣∣x=0

+x3

3!

d3y

dx3

∣∣∣∣x=0

+ . . . .

A binomial expansion

(1 + x)n = 1 + nx+n(n− 1)

2!x2 +

n(n− 1)(n− 2)

3!x3 + . . . .

If n is a positive integer the series terminates and is valid for all x: the term in xr is nCrxr or

(n

r

)xr, where(

n

r

)= nCr ≡ n!

r!(n−r)! is the number of different ways in which an unordered sample of r objects can be selected from

a set of n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series

is convergent for |x| < 1.

8

2.3 Power series expansions with real variables

ex = 1 + x+x2

2!+ . . .+

xn

n!+ . . . valid for all x,

ln(1 + x) = x− x2

2+x3

3+ . . .+ (−1)n+1x

n

n+ . . . valid for − 1 < x < 1,

cosx =eix + e−ix

2= 1− x2

2!+x4

4!− x6

6!+ . . . valid for all values of x,

sinx =eix − e−ix

2i= x− x3

3!+x5

5!+ . . . valid for all values of x,

tanx = x+1

3x3 +

2

15x5 + . . . valid for − π

2< x <

π

2,

tan−1 x = x− x3

3+x5

5− . . . valid for − 1 ≤ x ≤ 1,

sin−1 x = x+1

2

x3

3+

1

2

3

4

x5

5+ . . . valid for − 1 ≤ x ≤ 1,

coshx = 1 +x2

2!+x4

4!+ . . . valid for all x,

sinhx = x+x3

3!+x5

5!+ . . . valid for all x.

2.4 Integer series

N∑1

n = 1 + 2 + 3 + . . .+N =N(N − 1)

2,

N∑1

n2 = 12 + 22 + 32 + . . .+N2 =N(N + 1)(2N + 1)

6,

N∑1

n3 = 13 + 23 + 33 + . . .+N3 = [1 + 2 + 3 + . . .+N ]2

=N2(N + 1)2

4,

∞∑1

(−1)n+1

n= 1− 1

2+

1

3− 1

4+ · · · = ln 2,

∞∑1

(−1)n+1

2n− 1= 1− 1

3+

1

5− 1

7+ · · · = π

4,

∞∑1

1

n2= 1 +

1

4+

1

9+

1

16+ · · · = π2

6,

N∑1

n(n+ 1)(n+ 2) = 1.2.3 + 2.3.4 + · · ·+N(N + 1)(N + 2) =N(N + 1)(N + 2)(N + 3)

4,

N∑1

n(n+ 1)(n+ 2) . . . (n+ r) =N(N + 1)(N + 2) . . . (N + r)(N + r + 1)

r + 2.

9

2.5 Plane wave expansion

exp(ikz) = exp(ikr cos θ) =

∞∑l=0

(2l + 1)iljl(kr)Pl(cos θ),

where Pl(cos θ) are Legendre polynomials (see section 12.3) and jl(kr) are spherical Bessel functions, defined by

jl(kr) =√

π2ρJl+1/2(ρ), with jl(x) the Bessel function of order l (see section 12.4).

3 Vector Identities

If i, j,k are orthogonal unit vectors (orthonormal vectors) and a = axi+ ayj + azk, then |a|2 = a2x + a2

y + a2z.

3.1 Scalar product

a · b = |a| |b| cos θ = axbx + ayby + azbz = (ax, ay, az)

bx

by

bz

,

where θ is the angle between the vectors. Scalar multiplication is commutative: a · b = b · a.

3.2 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.

3.3 Equation of a plane

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

(a) r · d = |d| , where d is the unit normal from the origin to the plane, or

(b)x

X+y

Y+z

Z= 1, where X,Y, Z are the intercepts on the axes.

3.4 Vector product

The vector product has the form 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.

10

In a right handed system

a× b =

a2b3 − a3b2

a3b1 − a1b3

a1b2 − a2b1

.

Vector multiplication is not commutative : a× b = −b× a.

3.5 Scalar triplet product

(a× b) · c = a · (b× c) =

∣∣∣∣∣∣∣ax ay az

bx by bz

cx cy cz

∣∣∣∣∣∣∣ = −(a× c) · b, etc.

3.6 Vector triplet product

a× (b× c) = (a · c) b− (a · b) c, (a× b)× c = (a · c) b− (b · c) a.

3.7 Non-orthogonal basis

a = a1e1 + a2e2 + α3e3,

a1 = ε′ · a, where ε′ =e2 × e3

e1 . (e2 × e3)

and similarly for a2 and a3.

3.8 Summation convention

a = αiei implies summation over i = 1 . . . 3.

a · b = aibi,

(a× b)i = εijkajbk, where ε123 = 1; εijk = −εikj .

εijkεklm = δilδjm − δimδjl.

4 Matrix Algebra

4.1 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)ij = δij . If A is a square matrix of order n, then AI = IA = A. I is sometimes written as In if the order

needs to be stated explicitly and it should be clear that I = I−1.

11

4.2 Products

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

(AB)ij =

l∑k=1

AikBkj .

In general AB 6= BA.

4.3 Transpose matrices

If A is a matrix, then the transpose matrix AT is such that (AT )ij = (A)ji.

4.4 Determinants

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

|A| =∑i,j,k,...

εijk...A1iA2jA3k . . . ,

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

4.5 Inverse matrix

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

(A−1)ij =transpose of cofactor of Aij

|A|,

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

4.6 2× 2 matrices

If A =

(a b

c d

)then,

det A = |A| = ad− bc, AT =

(a c

b d

), A−1 =

1

|A|

(d −b−c a

).

4.7 Product rules

(AB . . .N)T = NT . . . BTAT ,

(AB . . .N)−1 = N−1 . . . B−1A−1,

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

12

4.8 Orthogonal matrices

An orthogonal matrix Q is a square matrix whose columns qi form a set of orthonormal vectors. For any orthogonal

matrix Q,

Q−1 = QT , |Q| = ±1, QT is also orthogonal.

4.9 Pauli spin matrices

σx =

(0 1

1 0

), σy =

(0 −ii 0

), σz =

(1 0

0 −1

),

σxσy = iσz, σyσz = iσx, σzσx = iσy, σxσx = σyσy = σzσz = I.

5 Vector Calculus

5.1 Notation

Define φ and is a scalar function of a set of position coordinates. In Cartesian coordinates we have φ = φ(x, y, z),

in cylindrical coordinates φ = φ(ρ, ϕ, z), and in spherical coordinates φ = φ(r, ϕ, θ), where the relationship between

Cartesian and cylindrical coordinates is given by

x = ρ cosϕ, y = ρ sinϕ, z = z,

and the relationship between Cartesian and spherical coordinates is given by

x = r cosϕ sin θ, y = r sinϕ sin θ, z = r cos θ.

Further, define a as a vector function whose components are scalar functions of the position coordinates a = iax + jay +

kaz, where ax, ay, az are independent functions of x, y, z.

In Cartesian coordinates

∇(“del”) ≡ i ∂∂x + j ∂∂y + k ∂

∂z ≡

∂∂x∂∂y∂∂z

.

grad φ = ∇φ, div a = ∇ · a, curl a = ∇× a.

13

5.2 Identities

With φ and ψ as scalar functions and a and b as vector fields (as defined above)

grad(φ+ ψ) ≡ grad φ+ grad ψ,

div(a+ b) ≡ div a+ div b,

grad(φψ) ≡ φ grad ψ + ψ grad φ,

curl(a+ b) ≡ curl a+ curl b,

div(φ a) ≡ a · grad φ+ φ diva,

curl(φ a) ≡ φ curl a− a× grad φ,

div(a× b) ≡ b · curl a− a · curl b,

curl(a× b) ≡ a div b− b div a+ (b · grad)a− (a · grad)b,

div(curl a) ≡ 0, curl(grad φ) ≡ 0,

curl curl a ≡ grad(div a)−∇2a,

grad(a · b) ≡ a× (curl b) + (a · grad)b+ b× (curl a) + (b · grad)a.

14

5.3 Grad, Div, Curl and the Laplacian in Cartesian, cylindrical and spherical coordinates.

Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates

Conversion to x = ρ cosϕ x = r cosϕ sin θ

Cartesian y = ρ sinϕ y = r sinϕ sin θ

Coordinates z = z z = r cos θ

Vector a axi+ ayj + azk aρρ+ aϕϕ+ azz arr + aθθ + aϕϕ

Gradient ∇φ∂φ

∂xi+

∂φ

∂yj +

∂φ

∂zk

∂φ

∂ρρ+

1

ρ

∂φ

∂ϕϕ+

∂φ

∂zz

∂φ

∂rr +

1

r

∂φ

∂θθ +

1

r sin θ

∂φ

∂ϕϕ

1

r2

∂(r2ar)

∂r+

1

sin θ

∂(aθ sin θ)

∂θDivergence∇ · a

∂ax∂x

+∂ay∂y

+∂az∂z

1

ρ

∂(ρaρ)

∂ρ+

1

ρ

∂aϕ∂ϕ

+∂az∂z

+1

r sin θ

∂aϕ∂ϕ

Curl ∇× a

∣∣∣∣∣∣∣∣∣i j k

∂

∂x

∂

∂y

∂

∂z

ax ay az

∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣

1

ρρ ϕ

1

ρz

∂

∂ρ

∂

∂ϕ

∂

∂z

aρ ρ aϕ az

∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣

1

r2 sin θr

1

r sin θθ

1

rϕ

∂

∂r

∂

∂θ

∂

∂ϕ

ar r aθ r aϕ sin θ

∣∣∣∣∣∣∣∣∣∣∣1

r2

∂

∂r

(r2 ∂φ

∂r

)+

1

r2 sin θ

∂

∂θ

(sin θ

∂φ

∂θ

)Laplacian

∇2φ∂2φ

∂x2+∂2φ

∂y2+∂2φ

∂z2

1

ρ

∂

∂ρ

(ρ∂φ

∂ρ

)+

1

ρ2

∂2φ

∂ϕ2+∂2φ

∂z2

+1

r2 sin2 θ

∂2φ

∂φ2

5.4 Transformation of integrals

Notation: l = the distance along some curve ‘C’ in space and is measured from some fixed point,

S = a surface area,

V = a volume contained by a specific surface,

t = the unit tangent to C at the point P ,

n = the unit outward pointing normal,

a = some vector function,

dl = the vector element of curve (= t dl),

dS = the vector element of surface (= n ds).

Then ∫C

a · t dl =

∫C

a · dl

and when a = ∇φ ∫C

(∇φ) · dl =

∫C

dφ.

15

5.4.1 Gauss’s theorem (divergence theorem)

When S defines a closed region having volume τ

∫τ

(∇ · a) dτ =

∫S

(a · n) dS =

∫S

a · dS.

Also ∫τ

(∇φ) dτ =

∫S

φdS

and ∫τ

(∇× a)dτ =

∫S

(n× a) dS.

5.4.2 Stoke’s theorem

When C is closed and bounds the open surface S,∫S

(∇× a) · dS =

∫C

a . dL

and also ∫S

(n×∇φ) dS =

∫C

φdL.

5.4.3 Green’s theorem

With φ and ψ as scalar functions,∫S

ψ∇φ · dS =

∫τ

∇ · (ψ∇φ) dτ =

∫τ

[ψ∇2φ+ (∇ψ) · (∇φ)

]dτ.

5.4.4 Green’s second theorem

∫τ

(ψ∇2φ− φ∇2ψ) dτ =

∫S

[ψ(∇φ)− φ(∇ψ)] . dS.

6 Complex Variables

6.1 Complex numbers

The complex number

z = x+ iy = r (cos θ + i sin θ) = rei(θ+2πn),

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 = r (cos θ − i sin θ) = re−i(θ+2πn); zz∗ = |z|2 = x2 + y2.

16

6.2 De Moivre’s theorem

(cos θ + i sin θ)n = einθ = cosnθ + i sinnθ.

6.2.1 Complex logarithm

log z = log |z|+ i arg z.

6.3 Cauchy’s residue theorem

Let γ be a simple closed positively oriented contour. If f(z) is a holomorphic function inside and on γ, except at finitely

many isolated singularities at z1, z2, . . . , zn inside γ, then∮γ

f(z) dz = 2πi

n∑j=1

res(f, zj).

6.4 Power series of complex variables

ez = 1 + z +z2

2!+ . . .+

zn

n!+ . . . , convergent for all finite z.

sin z = z − z3

3!+z5

5!− . . . , convergent for all finite z.

cos z = 1− z2

2!+z4

4!− . . . , convergent for all finite z.

principal value of ln(1 + z) andln(1 + z) = z − z2

2+z3

3− . . . , converges both on and within the circle

|z| = 1 except at the point z = −1.

converges both on and withintan−1 z = z − z3

3+z5

5− . . . , the circle |z| = 1 except at

the points z = ±i.

converges both on and within(1 + z)n = 1 + nz +

n(n− 1)

2!z2 +

n(n− 1)(n− 2)

3!z3 + . . . , the circle |z| = 1 except at

the point z = −1.

6.4.1 Laurent series

The Laurent series for a complex function f(z)

f(z) =

∞∑n=−∞

an (z − c)n ,

17

where an and c are constants, defined by the line integral

an =1

2πi

∮γ

f(w)

(w − c)n+1 dw,

where the path of integration γ is counterclockwise around a closed path enclosing c and lying in an annulus in which

f(z) is holomorphic.

7 Trigonometric Formulae

7.1 Identities

cos2 x+ sin2 x = 1, sec2 x− tan2 x = 1, cosec2 x− cot2 x = 1,

sin 2x = 2 sinx cosx, cos 2x = cos2 x− sin2 x, tan 2x =2 tanx

1− tan2 x,

sin(x± y) = sinx cos y ± cosx sin y, cosx cos y =cos(x+ y) + cos(x− y)

2,

cos(x± y) = cosx cos y ∓ sinx sin y, sinx sin y =cos(x− y)− cos(x+ y)

2,

tan(x± y) =tanx± tan y

1∓ tanx tan y, sinx cos y =

sin(x+ y) + sin(x− y)

2,

sinx+ sin y = 2 sinx+ y

2cos

x− y2

, cos2 x =1 + cos 2x

2,

sinx− sin y = 2 cosx+ y

2sin

x− y2

, sin2 x =1− cos 2x

2,

cosx+ cos y = 2 cosx+ y

2cos

x− y2

, cos3 x =3 cosx+ cos 3x

4,

cosx− cos y = −2 sinx+ y

2sin

x− y2

, sin3 x =3 sinx− sin 3x

4.

18

7.2 Relations between sides and angles of any plane triangle

In a plane triangle with angles x, y, and z and side opposite X,Y, and Z respectively,

X

sinx=

Y

sin y=

Z

sin z= diameter of circumscribed circle,

X2 = Y 2 + Z2 − 2Y Z cosx,

X = Y cos z + Z cosx,

cosx =Y 2 + Z2 − Z2

2Y Z,

tanx− y

2=X − YX + Y

cotz

2,

area =1

2XY sin z =

1

2Y Z sinx =

1

2ZX sin y =

√s(s−X)(s− Y )(s− Z), where s =

1

2(X + Y + Z).

7.3 Relations between sides and angles of any spherical triangle

In a spherical triangle with angles x, y, and z and slides opposite X,Y, and Z respectively,

sinX

sinx=

sinY

sin y=

sinZ

sin z,

cosX = cosY cosZ + sinY sinZ cosx,

cosx = − cos y cos z + sin y sin z cosX.

19

8 Hyperbolic Functions

coshx =1

2(ex + e−x),

sinhx =1

2(ex − e−x).

cosh ix = cosx, cos ix = coshx,

sinh ix = i sinx, sin ix = i sinhx,

tanhx =sinhx

coshx, sechx =

1

coshx,

cothx =coshx

sinhx, cosechx =

1

sinhx,

cosh2 x− sinh2 x = 1.

For large positive x:

coshx ≈ sinhx→ ex

2,

tanhx→ 1.

For large negative x:

coshx ≈ − sinhx→ e−x

2,

tanhx→ −1.

20

8.1 Relations of the functions

sinhx = − sinh(−x), sechx = sech(−x),

coshx = cosh(−x), cosechx = − cosech(−x),

tanhx = − tanh(−x), cothx = − coth(−x),

sinhx =2 tanh(x/2)

1− tanh2(x/2)=

tanhx√1− tanh2 x

, coshx =1 + tanh2(x/2)

1− tanh2(x/2)=

1√1− tanh2 x

,

tanh =√

1− sech2 x, sechx =√

1− tanh2 x,

cothx =√

cosech2 x+ 1, cosechx =√

coth2 x− 1,

sinh(x/2) =

√coshx− 1

2, cosh(x/2) =

√coshx+ 1

2,

tanh(x/2) =coshx− 1

sinhx=

sinhx

coshx+ 1,

sinh(2x) = 2 sinhx coshx, tanh(2x) =2 tanhx

1 + tanh2 x,

cosh(2x) = cosh2 x+ sinh2 x = 2 cosh2 x− 1 = 1 + 2 sinh2 x,

sinh(3x) = 3 sinhx+ 4 sinh3 x, cosh 3x = 4 cosh3 x− 3 coshx,

tanh(3x) =3 tanhx+ tanh3 x

1 + 3 tanh2 x,

sinh(x± y) = sinhx cosh y ± coshx sinh y, cosh(x± y) = coshx cosh y ± sinhx sinh y,

tanh(x± y) =tanhx± tanh y

1± tanhx tanh y,

sinhx+ sin y = 2 sinh1

2(x+ y) cosh

1

2(x− y), coshx+ cosh y = 2 cosh

1

2(x+ y) cosh

1

2(x− y),

sinhx− sin y = 2 cosh1

2(x+ y) sinh

1

2(x− y), coshx− cosh y = 2 sinh

1

2(x+ y) sinh

1

2(x− y),

sinhx± coshx =1± tanh(x/2)

1∓ tanh(x/2)= e±x, tanhx± tanh y =

sinh(x± y)

coshx cosh y,

coth± coth y = ± sinh(x± y)

sinhx sinh y.

21

8.2 Inverse functions

sinh−1 x

a= ln

(x+√x2 + a2

a

), for −∞ < x <∞.

cosh−1 x

a= ln

(x+√x2 − a2

a

), for x ≥ a.

tanh−1 x

a=

1

2ln

(a+ x

a− x

), for x2 < a2.

coth−1 x

a=

1

2ln

(x+ a

x− a

), for x2 > a2.

sech−1 x

a= ln

(a

x+

√a2

x2− 1

), for 0 < x ≤ a.

cosech−1 x

a= ln

(a

x+

√a2

x2+ 1

), for x 6= 0.

9 Limits

ncxn → 0 as n→∞ if |x| < 1 (any fixed c).

xn/n!→ 0 as n→∞ (any fixed c).

(1 + x/n)n → ex as n→∞.

x lnx→ 0 as x→ 0.

If f(a) = g(a) = 0 then limx→a

f(x)

g(x)=f ′(a)

g′(a)(L’Hopital’s rule).

10 Differentiation

dy

dz=dy

dx

dx

dz,

(uv)′ = u′v + uv′,(uv

)′=u′v − uv′

v2,

(uv)(n) = u(n)v + nu(n−1)v(1) + . . .+ nCru(n−r)v(r) + . . .+ uv(n), Leibniz Theorem,

where

nCr ≡

(n

r

)=

n!

r!(n− r)!.

22

d

dx(sinx) = cosx,

d

dx(sinhx) = coshx,

d

dx(cosx) = − sinx,

d

dx(coshx) = sinhx,

d

dx(tanx) = sec2 x,

d

dx(tanhx) = sech2 x,

d

dx(secx) = secx tanx,

d

dx(sechx) = − sechx tanhx,

d

dx(cotx) = − cosec2 x,

d

dx(cothx) = − cosech2 x,

d

dx(cosecx) = − cosecx cotx,

d

dx(cosechx) = − cosechx cothx.

11 Integration

11.1 Standard forms

∫xn dx =

xn+1

n+ 1+ c, for n 6= −1,∫

1

xdx = ln x+ c,

∫lnx dx = x(lnx− 1) + c,∫

eax dx =1

aeax + c,

∫xeax dx = eax

(x

a− 1

a2

)+ c,

∫x lnx dx =

x2

2

(lnx− 1

2

)+ c,∫

1

a2 + x2dx =

1

atan−1

(xa

)+ c,∫

1

a2 − x2dx =

1

atanh−1

(xa

)+ c =

1

2aln

(a+ x

a− x

)+ c, for x2 < a2,

∫1

x2 − a2dx = −1

acoth−1

(xa

)+ c =

1

2aln

(x− ax+ a

)+ c, for x2 > a2,∫

x

(x2 ± a2)ndx =

−1

2(n− 1)

1

(x2 ± a2)n−1+ c, for n 6= 1,∫

x

x2 ± a2dx =

1

2ln(x2 ± a2) + c,∫

1√a2 − x2

dx = sin−1(xa

)+ c.

23

∫1√

x2 ± a2dx = ln

(x+

√x2 ± a2

)+ c,∫

x√x2 ± a2

dx =√x2 ± a2 + c,∫ √

a2 − x2 dx =1

2

[x√a2 − x2 + a2 sin−1

(xa

)]+ c,∫ ∞

0

1

(1 + x)xpdx = π cosec pπ, for p < 1,

∫ ∞0

cos(x2) dx =

∫ ∞0

sin(x2) dx =1

2

√π

2,∫ ∞

−∞e−x

2/2σ2

dx = σ√

2π,

∫ ∞−∞

xne−x2/2σ2

dx =

1× 3× 5× . . . (n− 1)σn+1

√2π

0

for n ≥ 2 and even

for n ≥ 1 and odd,∫sinx dx = − cosx+ c,

∫sinhx dx = coshx+ c,∫

cosx dx = sinx+ c,

∫coshx dx = sinhx+ c,∫

tanx dx = − ln(cosx) + c,

∫tanhx dx = ln(coshx) + c,∫

cosecx dx = ln(cosecx− cotx) + c,

∫cosechx dx = ln [tanh(x/2)] + c,∫

secx dx = ln(secx+ tanx) + c,

∫sechx dx = 2 tan−1(ex) + c,∫

cotx dx = ln(sinx) + c,

∫cothx dx = ln(sinhx) + c,∫

sinmx sinnx dx =sin(m− n)x

2(m− n)− sin(m+ n)x

2(m+ n)+ c,

∫cosmx cosnx dx =

sin(m− n)x

2(m− n)+

sin(m+ n)x

2(m+ n)+ c,

11.2 Standard substitutions

If the integrand is a function of: substitute:

(a2 − x2) or√a2 − x2, x = a sin θ or x = a cos θ,

(x2 + a2) or√x2 + a2, x = a tan θ or x = a sinh θ,

(x2 − a2) or√x2 − a2, x = a sec θ or x = a cosh θ.

24

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

sinx =2t

1 + t2, cosx =

1− t2

1 + t2, dx =

2 dt

1 + t2.

If the integrand is of the form: substitute:∫dx

(ax+ b)√px+ q

, px+ q = u2,∫dx

(ax+ b)√px2 + qx+ r

, ax+ b =1

u.

11.3 Integration by parts

∫ b

a

u dv = uv

∣∣∣∣ba

−∫ b

a

v du.

11.4 Differentiation of an Integral

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

d

dα

∫ b(α)

a(α)

f(x, α) dx = f(b, α)db

dα− f(a, α)

da

dα+

∫ b(α)

a(α)

∂

∂αf(x, α) dx.

Special case,

d

dx

∫ x

a

f(y) dy = f(x).

11.5 Dirac δ− function

δ(t− τ) =1

2π

∫ ∞−∞

eiω(t−τ) dω.

If f(t) is an arbitrary function of t then ∫ ∞−∞

δ(t− τ)f(t) dt = f(τ).

δ(t) = 0 if t 6= 0, also ∫ ∞−∞

δ(t− τ) dt = 1.

The delta function satisfies the following scaling property for a non-zero scalar a,

25

∫ ∞−∞

δ(ax)dx =

∫ ∞−∞

δ(u)du

|a|=

1

|a|

and so

δ(ax) =δ(x)

|a|.

The delta distribution may be composed with a smooth function f(x) such that

δ(f(x)) =∑i

δ(x− xi)|f ′(xi)|

,

where the sum is over all roots of f(x), which are assumed to be simple.

11.6 Reduction formulae

11.6.1 Factorials

n! = n(n− 1)(n− 2) . . . 1, 0! = 1.

Stirling’s formula for large n:

ln(n!) ≈ n lnn− n.

For any p > −1, we can define the Gamma Function

Γ(p) =

∫ ∞0

xp−1e−xdx,

where for natural numbers (integers greater than 0)

n! = Γ(n+ 1) and Γ(n+ 1) = nΓ(n).

The values of the Gamma function at half-integer values are

Γ(1/2) = (−1/2)! =√π, Γ(3/2) = (1/2)! =

√π/2, etc. (11.1)

For any p, q > −1, ∫ 1

0

xp(1− x)q dx =p!q!

(p+ q + 1)!.

26

11.6.2 Trigonometrical

If m,n are integers,

∫ π/2

0

sinm θ cosn θ dθ =m− 1

m+ n

∫ π/2

0

sinm−2 θ cosn θ dθ =n− 1

m+ n

∫ π/2

0

sinm θ cosn−2 θ dθ

and can therefore be reduced eventually to one of the following integrals∫ π/2

0

sin θ cos θ dθ =1

2,

∫ π/2

0

sin θ dθ = 1,

∫ π/2

0

cos θ dθ = 1,

∫ π/2

0

dθ =π

2.

11.6.3 Other

If In =

∫ ∞0

xne−αx2

dx then In =(n− 1)

2αIn−2, I0 =

1

2

√π

α, I1 =

1

2α.

12 Differential Equations

12.1 Diffusion (conduction) equation

∂ψ

dt= κ∇2ψ.

12.2 Wave equation

∇2ψ =1

c2∂2ψ

∂t2.

12.3 Legendre’s equation

(1− x2)d2y

dx2− 2x

dy

dx+ l(l + 1)y = 0,

solutions of which are the Legendre polynomials Pl(x).

Rodrigues formula generates the polynomials

Pl(x) =1

2l l!

dl

dxl(x2 − 1)l.

which are P0(x) = 1, P1(x) = x, P2(x) = 12 (3x2 − 1), etc.

Recursion relation

Pl(x) =1

l[(2l − 1)xPl−1(x)− (l − 1)Pl−2(x)]

can also generate polynomials.

Orthogonality ∫ 1

−1

Pl(x)Pl′(x) dx =2

2l + 1δll′ .

27

12.4 Bessel’s equation

x2 d2y

dx2+ x

dy

dx+ (x2 −m2)y = 0.

solutions of which are Bessel functions Jm(x) of order m.

Series form of Bessel functions of the first kind

Jm(x) =

∞∑k=0

(−1)k (x/2)m+2k

k! (m+ k)!(integer m).

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

12.5 Laplace’s equation

∇2 u = 0.

If expressed in two-dimensional polar coordinates (see section 5), 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 5), a solution is

u(r, θ, ϕ) = [Arl +Br−(l+1)]Pml (cos θ) [C sinmϕ+D cosmϕ]

where l and m are integers with l ≥ |m| ≥ 0; A,B,C,D are constants;

Pml (cos θ) = sin|m| θ

[d

d(cos θ)

]|m|Pl(cos θ)

is the associated Legendre polynomial with P 0l (1) = 1.

If expressed in cylindrical polar coordinates (see section 5), a solution is

u(ρ, ϕ, z) = Jm(nρ)[A cosmϕ+B sinmϕ][C exp(nz) +D exp(−nz)],

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

12.6 Spherical harmonics

The normalised solutions Y ml (θ, ϕ) of the equation[1

sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

1

sin2 θ

∂2

∂ϕ2

]Y ml + l(l + 1)Y ml = 0

are called spherical harmonics, and have values given by

Y ml (θ, ϕ) =

√2l + 1

4π

(l − |m|)!(l + |m|)!

Pml (cos θ)eimϕ ×

(−1)m for m ≥ 0

1 for m < 0,

Y 00 =

√1

4π , Y 01 =

√3

4π cos θ, Y ±11 =

√3

8π sin θ e±iϕ, etc.

Orthogonality ∫4π

Y ∗ml Y m′

l′ δΩ = δll′ δmm′ .

28

13 Calculus of Variations

The condition for I =∫ baF (y, y′, x) dx to have a stationary value is ∂F

∂y = ddx

∂F∂y′ where y′ = dy

dx . This is the Euler-Lagrange

equation.

14 Functions of Several Variables

If φ = f(x, y, z, . . .) then ∂φ∂x implies differentiation with respect to x keeping y, z... constant

dφ =∂φ

∂xdx+

∂φ

∂ydy +

∂φ

∂zdz + . . . and δφ ≈ ∂φ

∂xδx+

∂φ

∂yδy +

∂φ

∂zδz + . . . ,

where x, y, z, . . . are independent variables. ∂φ∂x is also written as

(∂φ∂x

)y,...

or ∂φ∂x |y,... when the variable kept constant need

to to be stated explicitly.

If φ is a well-behaved function then ∂2φ∂x∂y = ∂2φ

∂y∂x etc.

If φ = f(x, y), (∂φ

∂x

)y

=1(∂x∂φ

)y

,

(∂φ

∂x

)y

(∂x

∂y

)φ

(∂y

∂φ

)x

= −1

14.1 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

φ(x, y) = φ(a+ u, b+ v) = φ(a, b) + u∂φ

∂x+ v

∂φ

∂y+

1

2!

(u2 ∂

2φ

∂x2+ 2uv

∂2φ

∂x∂y+ v2 ∂

2φ

∂y2

)+ . . . ,

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

Stationary points

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

∂y = 0. Unless ∂2φ∂x2 = ∂2φ

∂y2 = ∂2φ∂x ∂y = 0, the following conditions

determine whether it is a minimum, a maximum or a saddle point.

Minimum: ∂2φ∂x2 > 0, or ∂2φ

∂y2 > 0, and ∂2φ∂x2

∂2φ∂y2 >

(∂2φ∂x ∂y

)2

.

Maximum: ∂2φ∂x2 < 0, or ∂2φ

∂y2 < 0, and ∂2φ∂x2

∂2φ∂y2 >

(∂2φ∂x ∂y

)2

.

Saddle point: ∂2φ∂x2

∂2φ∂y2 <

(∂2φ∂x ∂y

)2

.

If ∂2φ∂x2 = ∂2φ

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

14.2 Changing variables: the chain rule

If φ = f(x, y, . . .) and the variables x, y, . . . are functions of independent variables u, v, . . . then

∂φ

∂u=∂φ

∂x

∂x

∂u+∂φ

∂y

∂y

∂u+ . . . ,

∂φ

∂v=∂φ

∂x

∂x

∂v+∂φ

∂y

∂y

∂v+ . . . .

29

14.3 Changing variables in integrals - Jacobians

For a change of variable y = Cx in the one dimensional integral of f(x) over x from x1 to x2∫ x2

x1

f(x) dx =

∫ y2=Cx2

y1=Cx1

f(y)dy

C.

Switching integration limits ∫ x2

x1

f(x) dx = −∫ x1

x2

f(x) dx.

If an area A in the x, y plane maps into an area A′ in the u, v plane then

∫A

f(x, y) dx dy =

∫A′f(u, v) J du dv, where J =

∣∣∣∣∣ ∂x∂u

∂x∂v

∂y∂u

∂y∂v

∣∣∣∣∣ .The Jacobian J can also be written as ∂(x,y)

∂(u,v) .

The corresponding formula for volume integrals is

∫V

f(x, y, z) dx dy dz =

∫V ′f(u, v, w) J du dv dw, where J =

∣∣∣∣∣∣∣∂x∂u

∂x∂v

∂x∂w

∂y∂u

∂y∂v

∂y∂w

∂z∂u

∂z∂v

∂z∂w

∣∣∣∣∣∣∣ .

15 Fourier Series and Transforms

15.1 Fourier series

If y(x) is a function defined in the range −π ≤ x ≤ π then

y(x) ≈ c0 +

M∑m=1

cm cosmx+

M ′∑m=1

sm sinmx,

where the coefficients are

c0 =1

2π

∫ π

−πy(x) dx,

cm =1

π

∫ π

−πy(x) cosmx dx (m = 1, ...,M),

sm =1

π

∫ π

−πy(x) sinmx dx (m = 1, ...,M ′),

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

30

15.2 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 cosmωt+∑

sm sinmωt,

where

c0 =ω

2π

∫ T

0

y(t) dt, cm =ω

π

∫ T

0

y(t) cosmωt dt, sm =ω

π

∫ T

0

y(x) sinmωt dt.

Variable x, range 0 ≤ x ≤ L,

y(x) ≈ c0 +∑

cm cos2πmx

L+∑

sm sin2πmx

L,

where

c0 =1

L

∫ L

0

y(x) dx, cm =2

L

∫ L

0

y(x) cos2πmL

πdx, sm =

2

L

∫ L

0

y(x) sin2πmL

πdx.

15.3 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 sines are

required in the Fourier series and sm = 2π

∫ π−π y(x) sinmxdx. If, in addition, y(x) is symmetric about x = π/2, then the

coefficients sm are given by sm = 0 (for m even), and sm = 4π

∫ π/20

y(x) sinmxdx (for m odd).

If y(x) is an even (symmetric) function [i.e., y(−x) = y(x) ] defined in the range −π ≤ x ≤ π, then only constant and

cosine terms are required in the Fourier series and cm = 2π

∫ π−π y(x) cosmxdx. If, in addition, y(x) is anti-symmetric

about x = π/2, then c0 = 0 and the coefficients cm are given by cm = 0 (for m even), cm = 4π

∫ π/20

y(x) cosmxdx. (for m

odd). [These results also apply to Fourier series with more general ranges provided appropriate changes are made to the

limits of integration.]

15.4 Complex form of Fourier series

If y(x) is a function defined in the range −π ≤ x ≤ π then

y(x) ≈∞∑−∞

Cm eimx, Cm =

∫ π

−πy(x) e−imxdx,

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 eimωt, where Cm =ω

2π

∫ T

0

y(t) e−imωt dt.

Variable x′, range 0 ≤ x′ ≤ L

y(x′) =

∞∑−∞

Cm ei2πmωx′, where Cm =

1

L

∫ L

0

y(x′) e−i2πmωx′dx′.

31

15.5 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 xn = nx/N

(where n = −(N − 1), . . . N) then

y(xn) = c0 + c1 cosxn + c2 cos 2xn + · · ·+ cN−1 cos(N − 1)xn + cN cosNxn

+ s1 sinxn + s2 sin 2xn + · · ·+ sN−1 sin(N − 1)xn + sN sinNxn,

where the coefficients are

c0 =1

2N

∑y(xn),

cm =1

N

∑y(xn) cosmxn (m = 1, 2, . . . , N − 1),

cN =1

2N

∑y(xn) cosNxn,

sm =1

N

∑y(xn) sinmxn (m = 1, 2, . . . , N − 1),

sN =1

2N

∑y(xn) sinNxn,

each summation being over 2N sampling points xn.

15.6 Fourier transforms

If y(x) is a function defined in the range −∞ ≤ x ≤ ∞ then the Fourier transform y(ω) is defined by the equations

y(t) =1

2π

∫ ∞−∞

y(ω) eiωt dω y(ω) =

∫ ∞−∞

y(t)e−iωtdt.

If ω is replaced by 2πf , where f is the frequency, this relationship becomes

y(t) =

∫ ∞−∞

y(f) ei2πft df, y(f) =

∫ ∞−∞

y(t) e−i2πft dt.

If y(t) is symmetric about t = 0 then

y(t) =1

π

∫ ∞0

y(ω) cosωt dω, y(ω) = 2

∫ ∞0

y(t) cosωtdt.

If y(t) is anti-symmetric about t = 0 then

y(t) =1

π

∫ ∞0

y(ω) sinωt dω, y(ω) = 2

∫ ∞0

y(t) sinωtdt.

15.6.1 Specific cases

Top hat y(t) =

a for |t| ≤ τ

0 for |t| > τ,y(ω) = 2a

sinωτ

ω.

32

Saw tooth y(t) =

a(1− |t|/τ) for |t| ≤ τ

0 for |t| > τ,y(ω) =

2a

ω2τ(1− cosωτ).

Gaussian y(t) = exp(−t2/t20), y(ω) = t0√π exp(−ω2t20/2).

Modulated function y(t) = f(t)eiω0t, y(ω) = f(ω − ω0).

Sampling function y(t) =

∞∑m=−∞

δ(t−mτ), y(ω) =2π

τ

∞∑m=−∞

δ(ω − 2πn/τ).

15.7 Convolution theorem

z(t) =

∫ ∞−∞

x(τ) y(t− τ) dτ =

∫ ∞−∞

x(t− τ) y(τ) dτ = x(t) ∗ y(t), then z(ω) = x(ω) y(ω).

Conversely xy = x ∗ y/2π.

15.8 Parseval’s theorem∫ ∞−∞

y∗(t) y(t) dt =1

2π

∫ ∞−∞

y∗(ω) y(ω) dω, (if y normalised).

15.9 Fourier transforms in two dimensions

V (k) =

∫V (r) e−ik.r d2r

=

∫ ∞0

2π r V (r) J0(kr) dr, if azimuthally symmetric.

15.10 Fourier transforms in three dimensions

V (k) =

∫V (r) e−ik.r d3r

=4π

k

∫ ∞0

V (r) r sin kr dr, if spherically symmetric,

V (r) =1

(2π)3

∫V (k) eik.r d3k.

33

16 Numerical Analysis

16.1 Finding the zeros of equations

If the equation is y = f(x) and xn is an approximation to the root then either

xn+1 = xn −f(xn)

f ′(xn)(Newton)

or, xn+1 = xn −xn − xn−1

f(xn)− f(xn−1)f(xn) (Linear interpolation)

are, in general, better approximations.

16.2 Numerical integration of differential equations

Ifdy

dx= f(x, y) then

yn+1 = yn + hf(xn, yn) where h = xn+1 − xn (Euler method).

Putting y∗n+1 = yn + h f(xn, yn) (improved Euler method),

then yn+1 = yn +h [f(xn, yn) + f(xn+1, y

∗n+1])

2.

16.3 Central difference notation

If y(x) is tabulated at equal intervals of x, where h is the interval, then δyn+1/2 = yn+1 − yn and δ2yn = δyn+1/2 − δyn−1/2.

16.4 Approximating to derivatives

(dy

dx

)n

≈ yn+1 − ynh

≈ yn − yn−1

h≈yn+1/2 + yn−1/2

2h, where h = xn+1 − xn.

(d2y

dx2

)n

≈ yn+1 − 2yn + yn−1

h2=δ2ynh2

.

Interpolation: Everett’s formula

y(x) = y(x0 + θh) ≈ θ y0 + θ y1 +1

3!θ (θ2 − 1) δ2y0 +

1

3!θ (θ2 − 1) δ2y1 + . . . ,

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

represent linear interpolation.

34

16.5 Numerical evaluation of definite integrals

16.5.1 Trapezoidal rule

The interval of integration is divided into n equal sub-intervals, each of width h; then∫ b

a

f(x) dx ≈ h[

1

2f(a) + f(x1) + . . .+ f(xj) + · · ·+ 1

2f(b)

],

where h = (b− a)/n and xj = a+ jh.

16.5.2 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 ∫ b

a

f(x) dx ≈ h

3[f(a) + 4f(x1) + 2f(x2) + 4f(x3) + · · ·+ 2f(x2n−2) + 4f(x2n−1) + f(b)] .

16.5.3 Gauss’s integration formulae

These have the general form

∫ 1

−1

y(x) dx ≈n∑i=1

ci y(xi).

For n = 2 : xi = ±0.5773; ci = 1, 1 (exact for any cubic).

For n = 3 : xi = −0.7746, 0.0, 0.7746; ci = 0.555, 0.888, 0.555 (exact for any quintic).

17 Statistics and Treatment of Random Errors

17.1 Mean, variance and covariance

A random variable x has a distribution over some subset of the real numbers. When the distribution of x is discrete,

the probability to observe a value xi is Pi. When the distribution is continuous, the probability that x lies in an interval

[x, x+ dx] is f(x) dx, where f(x) is the probability density function. The mean µ and variance σ2 of x are defined as

µ = E[x] =∑i

Pi xi or

∫x f(x) dx ,

σ2 = E[(x− µ)2] =∑i

Pi (xi − µ)2 or

∫(x− µ)2 f(x) dx .

Given a sample of independent values x1, . . . , xn, the mean and variance can be estimated using the sample mean x and

sample variance s2, given by

35

x =1

n

n∑i=1

xi ,

s2 =1

n− 1

n∑i=1

(xi − x)2 .

The standard deviation σ (the square root of the variance) can be estimated by s =√s2. The standard deviation of the

sample mean is

σx =σ√n,

where σ is the standard deviation of the random variable x itself and n is the number of observed values in the sample

x1, . . . , xn. The sample mean σx can be estimated using sx = s/√n.

If z = ax+ by then E[z] = aE[x] + bE[y]. If x and y are independent then V [y] = a2V [x] + b2V [y].

For any pair of random variables x and y, the covariance cov[x, y] and the correlation coefficient ρxy are defined as

cov[x, y] = E[xy]− E[x]E[y] = E[(x− E[x])(y − E[y])] ,

ρxy =cov[x, y]

σxσy,

where σx and σy are the standard deviations of x and y, respectively. The correlation coefficient is dimensionless and lies

in the range −1 ≤ ρxy ≤ 1. For uncorrelated variables one has ρxy = 0. The covariance of any variable with itself is that

variable’s variance, i.e., cov[x, x] = σ2x, and thus ρxx = 1. For three random variables x, y and z and constants a, b and

c, cov[ax+ by, cz] = accov[x, z] + bccov[y, z].

If one has a set of n independent pairs of values (x1, y1), . . . , (xn, yn), then these can be used to construct estimators

(written here with hats) for the covariance and correlation coefficient using

cov[x, y] =1

n− 1

n∑i=1

(xi − x)(yi − y) =n

n− 1(xy − x y) ,

ρxy =cov[x, y]

sxsy=

xy − x y√(x2 − x2)(y2 − y2)

.

17.2 Error propagation

Consider a set of n measured quantities x = (x1, . . . , xn) with a given covariance matrix Vij = cov[xi, xj ]. Suppose from

these one constructs a set of m functions y(x) = (y1(x), . . . , ym(x)). Using error propagation, the covariance matrix of

the functions Uij = cov[yi, yj ] can be approximated as

36

cov[yi, yj ] =

n∑k,l=1

∂yi∂xk

∂yj∂xl

Vkl ,

where the derivatives are evaluated at the observed values of the variables x. The approximation is based on a first-order

Taylor expansion of the functions y(x) and is thus exact if the functions are linear.

If the quantities x1, . . . xn are independent (and hence uncorrelated), then their covariance matrix is diagonal, i.e.,

Vij = σ2i δij , where σi is the standard deviation of xi. In this case, the error propagation formula becomes

cov[yi, yj ] =

n∑k=1

∂yi∂xk

∂yj∂xk

σ2k .

If there is only one function y(x1, . . . , xn), then the formula for its variance becomes

σ2y =

n∑k=1

(∂y

∂xk

)2

σ2k .

As special cases, if y(x) = ax1 +bx2, where a and b are constants and x1 and x2 are uncorrelated with standard deviations

σ1 and σ2, then the standard deviation of y is

σy =√a2σ2

1 + b2σ22 .

If y = xa1xb2, then σy/y (the relative error) is found to be

σyy

=

√a2σ2

1

x21

+ b2σ2

2

x22

.

In both of the common special cases y = x1x2 and y = x1/x2 one has a2 = b2 = 1.

For y = ex, error propagation gives σy/y = x; for y = lnx one finds σy = σx/x.

17.3 Probability distributions

Binomial: P (k) = n!k!(n−k)! p

k (1− p)n−k, k = 0, 1, . . . , n, E[k] = np, V [k] = np(1− p), 0 ≤ p ≤ 1.

Poisson: P (k) = µk

k! e−µ, k = 0, 1, . . . , E[k] = µ, V [k] = µ, µ ≥ 0.

Gaussian: f(x) = 1√2πσ

exp[−(x−µ)2

2σ2

], −∞ < x <∞, E[x] = µ, V [x] = σ2.

Chi-square: f(x) = 12n/2Γ(n/2)

xn/2−1e−x/2, x ≥ 0, n = num. deg. of freedom = 1, 2, . . . , E[x] = n, V [x] = 2n.

37

17.4 Method of least squares

Consider measurements yi with i = 1, . . . , N , that are treated as independent random variables with given standard

deviations σi. Suppose the expectation values of the yi are given by a function f(x;θ), i.e., E[yi] = f(xi;θ), where

θ = (θ1, . . . , θm) is a set of m parameters, whose values we want to estimate using the measured data.

To find the least-squares estimators θ of the parameters θ, one minimises the quantity

χ2(θ) =

N∑i=1

(yi − f(xi;θ))2

σ2i

.

The statistical errors on the fitted parameters are found from the matrix of second derivatives of χ2(θ) evaluated at its

minimum. In general the estimators are correlated, with covariances

cov[θi, θj ] =1

2

(∂2χ2

∂θi∂θj

)θ=θ

.

If the hypothesised functional form of f(x;θ) is correct, then the minimised value of χ2 should follow a chi-square

distribution with a number of degrees of freedom equal to the number of measured values minus the number of fitted

parameters (n = N −m).

If the measurements y1, . . . , yN are correlated and have a given covariance matrix Vij = cov[yi, yj ], then the formula for

χ2(θ) becomes

χ2(θ) =

N∑i,j=1

(yi − f(xi;θ))(V −1)ij(yj − f(xj ;θ)) ,

where V −1 is the inverse of the covariance matrix V .

38