University of Manchester MATHEMATICAL FORMULA TABLES

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University of Manchester MATHEMATICAL FORMULA TABLES Version 2.1

November 2004

Other than this front cover these tables are identical to the UMIST, version 2.0 tables.

1

CONTENTS page Greek Alphabet

3

Indices and Logarithms

3

Trigonometric Identities

4

Complex Numbers

6

Hyperbolic Identities

6

Series

7

Derivatives

9

Integrals

11

Laplace Transforms

13

Z Transforms

16

Fourier Series and Transforms

17

Numerical Formulae

19

Vector Formulae

23

Mechanics

25

Algebraic Structures

27

Statistical Distributions

29

F - Distribution

29

Normal Distribution

31

t - Distribution

32

χ2 (Chi-squared) - Distribution

33

Physical and Astronomical constants

34

2

GREEK ALPHABET Aα

alpha



nu



beta

Ξξ

xi

Γγ

gamma

Oo

omicron

∆δ

delta

Ππ

pi

E ², ε

epsilon



rho



zeta

Σσ

sigma



eta



tau

Θ θ, ϑ theta

Υυ

upsilon



iota

Φ φ, ϕ

phi



kappa



chi

Λλ

lambda

Ψψ

psi



mu

Ωω

omega

INDICES AND LOGARITHMS

am × an = am+n (am )n = amn log(AB) = log A + log B log(A/B) = log A − log B log(An ) = n log A logc a logb a = logc b

3

TRIGONOMETRIC IDENTITIES

tan A = sin A/ cos A sec A = 1/ cos A cosec A = 1/ sin A cot A = cos A/ sin A = 1/ tan A sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec 2 A = 1 + cot2 A sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B ∓ sin A sin B tan(A ± B) =

tan A ± tan B 1 ∓ tan A tan B

sin 2A = 2 sin A cos A cos 2A = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A tan 2A =

2 tan A 1 − tan2 A

sin 3A = 3 sin A − 4 sin3 A cos 3A = 4 cos3 A − 3 cos A tan 3A =

3 tan A − tan3 A 1 − 3 tan2 A

cos A−B sin A + sin B = 2 sin A+B 2 2

4

sin A−B sin A − sin B = 2 cos A+B 2 2 cos A−B cos A + cos B = 2 cos A+B 2 2 cos A − cos B = −2 sin A+B sin A−B 2 2 2 sin A cos B = sin(A + B) + sin(A − B) 2 cos A sin B = sin(A + B) − sin(A − B) 2 cos A cos B = cos(A + B) + cos(A − B) −2 sin A sin B = cos(A + B) − cos(A − B) a sin x + b cos x = R sin(x + φ), where R = If t = tan cos x =

1 2

1 x 2



a2 + b2 and cos φ = a/R, sin φ = b/R.

1 − t2 2t , cos x = . then sin x = 1 + t2 1 + t2

(eix + e−ix ) ;

eix = cos x + i sin x ;

sin x =

1 2i

(eix − e−ix )

e−ix = cos x − i sin x

5

COMPLEX NUMBERS

i=



−1

Note:- ‘j’ often used rather than ‘i’.

Exponential Notation

eiθ = cos θ + i sin θ

De Moivre’s theorem [r(cos θ + i sin θ)]n = rn (cos nθ + i sin nθ) nth roots of complex numbers If z = reiθ = r(cos θ + i sin θ) then z 1/n =

√ n

rei(θ+2kπ)/n ,

k = 0, ±1, ±2, ...

HYPERBOLIC IDENTITIES cosh x = (ex + e−x ) /2

sinh x = (ex − e−x ) /2

tanh x = sinh x/ cosh x sechx = 1/ cosh x

cosechx = 1/ sinh x

coth x = cosh x/ sinh x = 1/ tanh x cosh ix = cos x

sinh ix = i sin x

cos ix = cosh x

sin ix = i sinh x cosh2 A − sinh2 A = 1 sech2 A = 1 − tanh2 A cosech 2 A = coth2 A − 1

6

SERIES

Powers of Natural Numbers n X

n n X X 1 1 1 2 k = n(n + 1) ; k = n(n + 1)(2n + 1); k 3 = n2 (n + 1)2 2 6 4 k=1 k=1 k=1

Arithmetic

Sn =

n−1 X

(a + kd) =

k=0

n {2a + (n − 1)d} 2

Geometric (convergent for −1 < r < 1) Sn =

n−1 X

ark =

k=0

a(1 − rn ) a , S∞ = 1−r 1−r

Binomial (convergent for |x| < 1)

à !

à !

n 2 n r (1 + x) = 1 + nx + x + ... + x + ... 2 r n

where

à !

n n(n − 1)(n − 2)...(n − r + 1) = r! r

Maclaurin series f (x) = f (0) + xf 0 (0) + where Rk+1 =

x2 00 xk f (0) + ... + f (k) (0) + Rk+1 2! k!

xk+1 (k+1) f (θx), 0 < θ < 1 (k + 1)!

Taylor series f (a + h) = f (a) + hf 0 (a) + where Rk+1 =

h2 00 hk f (a) + ... + f (k) (a) + Rk+1 2! k!

hk+1 (k+1) f (a + θh) , 0 < θ < 1. (k + 1)!

OR f (x) = f (x0 ) + (x − x0 )f 0 (x0 ) + where Rk+1 =

(x − x0 )2 00 (x − x0 )k (k) f (x0 ) + ... + f (x0 ) + Rk+1 2! k!

(x − x0 )k+1 (k+1) f (x0 + (x − x0 )θ), 0 < θ < 1 (k + 1)! 7

Special Power Series

xr x2 x3 + + ... + + ... 2! 3! r!

ex = 1 + x +

(all x)

sin x = x −

x3 x5 x7 (−1)r x2r+1 + − + ... + + ... 3! 5! 7! (2r + 1)!

(all x)

cos x = 1 −

x2 x4 x6 (−1)r x2r + − + ... + + ... 2! 4! 6! (2r)!

(all x)

tan x = x +

x3 2x5 17x7 + + + ... 3 15 315

(|x| < π2 )

1 x3 1.3 x5 1.3.5 x7 + + + 2 3 2.4 5 2.4.6 7

sin−1 x = x +

... +

tan−1 x = x −

1.3.5....(2n − 1) x2n+1 + ... 2.4.6....(2n) 2n + 1

x3 x5 x7 x2n+1 + − + ... + (−1)n + ... 3 5 7 2n + 1

`n(1 + x) = x −

x2 x3 x4 xn + − + ... + (−1)n+1 + ... 2 3 4 n

(|x| < 1)

(|x| < 1)

(−1 < x ≤ 1)

sinh x = x +

x3 x5 x7 x2n+1 + + + ... + + ... 3! 5! 7! (2n + 1)!

(all x)

cosh x = 1 +

x2 x4 x6 x2n + + + ... + + ... 2! 4! 6! (2n)!

(all x)

tanh x = x −

x3 2x5 17x7 + − + ... 3 15 315

(|x| < π2 )

sinh−1 x = x −

1 x3 1.3 x5 1.3.5 x7 + − + 2 3 2.4 5 2.4.6 7

... + (−1)n tanh−1 x = x +

1.3.5...(2n − 1) x2n+1 + ... 2.4.6...2n 2n + 1

x3 x5 x7 x2n+1 + + + ... + ... 3 5 7 2n + 1 8

(|x| < 1)

(|x| < 1)

DERIVATIVES function

derivative

xn

nxn−1

ex

ex

ax (a > 0)

ax `na

`nx

1 x

loga x

1 x`na

sin x

cos x

cos x

− sin x

tan x

sec2 x

cosec x

− cosec x cot x

sec x

sec x tan x

sin−1 x

− cosec 2 x 1 √ 1 − x2

cos−1 x

−√

cot x

1 1 − x2

sinh x

1 1 + x2 cosh x

cosh x

sinh x

tanh x

sech 2 x

cosech x

− cosech x coth x

tan−1 x

sech x

− sech x tanh x

sinh−1 x

− cosech2 x 1 √ 1 + x2

cosh−1 x(x > 1)



tanh−1 x(|x| < 1)

1 1 − x2

coth−1 x(|x| > 1)



coth x

9

1 x2 − 1

x2

1 −1

Product Rule d dv du (u(x) v(x)) = u(x) + v(x) dx dx dx Quotient Rule d dx

Ã

u(x) v(x)

!

=

dv v(x) du − u(x) dx dx [v(x)]2

Chain Rule d (f (g(x))) = f 0 (g(x)) × g 0 (x) dx Leibnitz’s theorem à !

dn n(n − 1) (n−2) (2) n (n−r) (r) f .g +...+f.g (n) (f.g) = f (n) .g +nf (n−1) .g (1) + f .g +...+ n dx 2! r

10

INTEGRALS function dg(x) f (x) dx xn (n 6= −1)

integral

e

ex

sin x

− cos x

1 x x

cos x

f (x)g(x) − xn+1 n+1

Z

df (x) g(x)dx dx

`n|x|

Note:- `n|x| + K = `n|x/x0 |

sin x

tan x

`n| sec x|

cosec x

−`n| cosec x + cot x|

sec x

¯ ¯

or

`n| sec x + tan x| = `n ¯tan

³

π 4

+

cot x 1 2 a + x2

`n| sin x| 1 x tan−1 a a

1 a2 − x 2

1 a+x `n 2a a − x

or

1 x tanh−1 a a

1 − a2

1 x−a `n 2a x + a

or



x2

1 − x2

sin−1

x a

a2



1 2 a + x2

sinh−1

x a

or

`n x +

1 x2 − a2 sinh x

cosh−1

x a

or

`n|x +

cosh x

cosh x

sinh x

tanh x

`n cosh x

cosech x

−`n |cosech x + coth x|

sech x coth x

¯

x 2

´¯ ¯ ¯

(|x| < a)

1 x coth−1 a a





¯

`n ¯¯tan x2 ¯¯

(|x| > a)

(a > |x|)

2 tan−1 ex

`n| sinh x|

11

³

´ √ x2 + a 2



x2 − a2 |

or

(|x| > a)

¯

¯

`n ¯¯tanh x2 ¯¯

Double integral

where

Z Z

f (x, y) dx dy =

Z Z

¯ ¯ ∂(x, y) ¯¯ J= = ¯¯ ∂(r, s) ¯

12

g(r, s) J dr ds

∂x ∂r ∂y ∂r

∂x ∂s ∂y ∂s

¯ ¯ ¯ ¯ ¯ ¯ ¯

LAPLACE TRANSFORMS

function

R f˜(s) = 0∞ e−st f (t)dt

transform 1 s n! n+1 s 1 s−a ω s2 + ω 2 s s2 + ω 2 ω 2 s − ω2 s s2 − ω 2

1 tn eat sin ωt cos ωt sinh ωt cosh ωt t sin ωt

2ωs (s2 + ω 2 )2

t cos ωt

s2 − ω 2 (s2 + ω 2 )2

Ha (t) = H(t − a)

e−as s

δ(t)

1

eat tn

n! (s − a)n+1

eat sin ωt

ω (s − a)2 + ω 2

eat cos ωt

s−a (s − a)2 + ω 2

eat sinh ωt

ω (s − a)2 − ω 2

eat cosh ωt

s−a (s − a)2 − ω 2 13

Let f˜(s) = L {f (t)} then n

o

= f˜(s − a),

)

=

L eat f (t)

L {tf (t)} = − (

f (t) L t

Z

d ˜ (f (s)), ds



f˜(x)dx if this exists.

x=s

Derivatives and integrals Let y = y(t) and let y˜ = L {y(t)} then (

)

dy L = s˜ y − y0 , dt ) ( d2 y = s2 y˜ − sy0 − y00 , L dt2 ½Z t ¾ 1 L y˜ y(τ )dτ = s τ =0 where y0 and y00 are the values of y and dy/dt respectively at t = 0. Time delay Let then

   0

t
g(t) = Ha (t)f (t − a) = 

 f (t − a)

t>a

L {g(t)} = e−as f˜(s).

Scale change µ ¶

s 1 . L {f (kt)} = f˜ k k Periodic functions Let f (t) be of period T then L {f (t)} =

Z T 1 e−st f (t)dt. 1 − e−sT t=0

14

Convolution Let then

f (t) ∗ g(t) =

Rt

x=0

f (x)g(t − x)dx =

L {f (t) ∗ g(t)} = f˜(s)˜ g (s).

Rt

x=0

f (t − x)g(x)dx

RLC circuit For a simple RLC circuit with initial charge q0 and initial current i0 , µ



1 1 e i − Li0 + E˜ = r + Ls + q0 . Cs Cs Limiting values initial value theorem lim f (t) = s→∞ lim sf˜(s),

t→0+

final value theorem lim f (t) =

Z

t→∞ ∞ 0

f (t)dt =

provided these limits exist.

15

lim sf˜(s),

s→0+

lim f˜(s)

s→0+

Z TRANSFORMS

Z {f (t)} = f˜(z) =

∞ X

f (kT )z −k

k=0

function

transform

δt,nT e−at

z −n (n ≥ 0) z z − e−aT

te−at

T ze−aT (z − e−aT )2

t2 e−at

T 2 ze−aT (z + e−aT ) (z − e−aT )3

sinh at cosh at

z2

z sinh aT − 2z cosh aT + 1

z2

z(z − cosh aT ) − 2z cosh aT + 1

e−at sin ωt

ze−aT sin ωT z 2 − 2ze−aT cos ωT + e−2aT

e−at cos ωt

z(z − e−aT cos ωT ) z 2 − 2ze−aT cos ωT + e−2aT

te−at sin ωt

T ze−aT (z 2 − e−2aT ) sin ωT (z 2 − 2ze−aT cos ωT + e−2aT )2

te−at cos ωt

T ze−aT (z 2 cos ωT − 2ze−aT + e−2aT cos ωT ) (z 2 − 2ze−aT cos ωT + e−2aT )2

Shift Theorem P n−k f (kT ) (n > 0) Z {f (t + nT )} = z n f˜(z) − n−1 k=0 z Initial value theorem f (0) = limz→∞ f˜(z) 16

Final value theorem h

f (∞) = lim (z − 1)f˜(z) z→1

i

provided f (∞) exists.

Inverse Formula 1 Z π ikθ ˜ iθ f (kT ) = e f (e )dθ 2π −π

FOURIER SERIES AND TRANSFORMS

Fourier series ∞ X 1 {an cos nωt + bn sin nωt} f (t) = a0 + 2 n=1

where

2 Z t0 +T f (t) cos nωt dt T t0 2 Z t0 +T = f (t) sin nωt dt T t0

an = bn

17

(period T = 2π/ω)

Half range Fourier series

sine series

4 Z T /2 an = 0, bn = f (t) sin nωt dt T 0

cosine series

4 Z T /2 bn = 0, an = f (t) cos nωt dt T 0

Finite Fourier transforms sine transform

f˜s (n) = f (t) =

4 Z T /2 f (t) sin nωt dt T 0

∞ X

f˜s (n) sin nωt

n=1

cosine transform

4 Z T /2 f (t) cos nωt dt T 0 ∞ X 1˜ fc (0) + f˜c (n) cos nωt f (t) = 2 n=1

f˜c (n) =

Fourier integral µ ¶ 1 Z ∞ iωt Z ∞ 1 f (u)e−iωu du dω lim f (t) + lim f (t) = e t&0 2 t%0 2π −∞ −∞

Fourier integral transform

1 Z ∞ −iωu f˜(ω) = F {f (t)} = √ e f (u) du 2π −∞ o n 1 Z ∞ iωt ˜ −1 ˜ e f (ω) dω f (t) = F f (ω) = √ 2π −∞

18

NUMERICAL FORMULAE

Iteration Newton Raphson method for refining an approximate root x0 of f (x) = 0 xn+1 = xn − Particular case to find



Secant Method xn+1

f (xn ) f 0 (xn )

N use xn+1 =

1 2

Ã

³

xn +

f (xn ) − f (xn−1 ) = xn − f (xn )/ xn − xn−1

N xn

´

.

!

Interpolation

∆fn = fn+1 − fn , δfn = fn+ 1 − fn− 1 2 2 ´ 1³ ∇fn = fn − fn−1 , µfn = fn+ 1 + fn− 1 2 2 2 Gregory Newton Formula à !

p r p(p − 1) 2 fp = f0 + p∆f0 + ∆ f0 + ... + ∆ f0 2! r where p =

x − x0 h

Lagrange’s Formula for n points y=

n X

yi `i (x)

i=1

where `i (x) =

Πnj=1,j6=i (x − xj ) Πnj=1,j6=i (xi − xj )

19

Numerical differentiation Derivatives at a tabular point 1 1 hf00 = µ δf0 − µ δ 3 f0 + µ δ 5 f0 − ... 6 30 1 1 h2 f000 = δ 2 f0 − δ 4 f0 + δ 6 f0 − ... 12 90 1 1 1 1 hf00 = ∆f0 − ∆2 f0 + ∆3 f0 − ∆4 f0 + ∆5 f0 − ... 2 3 4 5 11 5 h2 f000 = ∆2 f0 − ∆3 f0 + ∆4 f0 − ∆5 f0 + ... 12 6 Numerical Integration Z

Trapezium Rule where

fi = f (x0 + ih), E = −

x0 +h x0

f (x)dx '

h (f0 + f1 ) + E 2

h3 00 f (a), x0 < a < x0 + h 12

Composite Trapezium Rule Z

x0 +nh x0

f (x)dx '

h h2 h4 000 {f0 + 2f1 + 2f2 + ...2fn−1 + fn } − (fn0 − f00 ) + (f − f0000 )... 2 12 720 n where f00 = f 0 (x0 ), fn0 = f 0 (x0 + nh), etc Z

Simpson’s Rule

x0 +2h x0

f (x)dx '

h5 (4) E = − f (a) 90

where

h (f0 +4f1 +f2 )+E 3

x0 < a < x0 + 2h.

Composite Simpson’s Rule (n even) Z

x0 +nh x0

f (x)dx '

where

h (f0 + 4f1 + 2f2 + 4f3 + 2f4 + ... + 2fn−2 + 4fn−1 + fn ) + E 3 E=−

20

nh5 (4) f (a). 180

x0 < a < x0 + nh

Gauss order 1. (Midpoint) Z

1 −1

f (x)dx = 2f (0) + E 2 00 E = f (a). 3

where Gauss order 2.

Z

!

Ã

1 f (x)dx = f − √ + f −1 3 1

where

E=

Ã

−1
!

1 √ +E 3

1 0v f (a). 135

−1
Differential Equations To solve y 0 = f (x, y) given initial condition y0 at x0 , xn = x0 + nh. Euler’s forward method yn+1 = yn + hf (xn , yn )

n = 0, 1, 2, ...

Euler’s backward method yn+1 = yn + hf (xn+1 , yn+1 )

n = 0, 1, 2, ...

Heun’s method (Runge Kutta order 2) h yn+1 = yn + (f (xn , yn ) + f (xn + h, yn + hf (xn , yn ))). 2 Runge Kutta order 4. h yn+1 = yn + (K1 + 2K2 + 2K3 + K4 ) 6 where K1 = f (xn , yn ) Ã ! h hK1 K2 = f xn + , y n + 2 2 Ã ! h hK2 K3 = f xn + , y n + 2 2 K4 = f (xn + h, yn + hK3 )

21

Chebyshev Polynomials h

Tn (x) = cos n(cos−1 x) To (x) = 1 Un−1 (x) =

i

T1 (x) = x

Tn0 (x) sin [n(cos−1 x)] √ = n 1 − x2

Tm (Tn (x)) = Tmn (x). Tn+1 (x) = 2xTn (x) − Tn−1 (x) Z

Un+1 (x) = 2xUn (x) − Un−1 (x) ( ) 1 Tn+1 (x) Tn−1 (x) Tn (x)dx = − + constant, 2 n+1 n−1

where and

R

1 f (x) = a0 T0 (x) + a1 T1 (x)...aj Tj (x) + ... 2 2Zπ f (cos θ) cos jθdθ aj = π 0

f (x)dx = constant +A1 T1 (x) + A2 T2 (x) + ...Aj Tj (x) + ...

where Aj = (aj−1 − aj+1 )/2j

j≥1

22

n≥2

j≥0

VECTOR FORMULAE

Scalar product a.b = ab cos θ = a1 b1 + a2 b2 + a3 b3 ¯ ¯ ¯ i ¯ ¯ ˆ = ¯¯ a1 Vector product a × b = ab sin θ n ¯ ¯ ¯ b1

j

¯

¯ k ¯¯

¯ a2 a3 ¯¯ ¯

¯ b2 b3 ¯

= (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k

Triple products ¯ ¯ ¯ a1 ¯ ¯ [a, b, c] = (a × b).c = a.(b × c) = ¯¯ b1 ¯ ¯ ¯ c1

a × (b × c) = (a.c)b − (a.b)c

Vector Calculus ∇ ≡

Ã

∂ ∂ ∂ , , ∂x ∂y ∂z

¯ ¯

a2 a3 ¯¯ ¯ b2 b3 ¯¯

!

grad φ ≡ ∇φ, div A ≡ ∇.A, curl A ≡ ∇ × A div grad φ ≡ ∇.(∇ φ) ≡ ∇2 φ (for scalars only) div curl A = 0

curl grad φ ≡ 0

∇2 A = grad div A − curl curl A ∇(αβ) = α ∇β + β ∇α div (αA) = α div A + A.(∇α) curl (αA) = α curl A − A × (∇α) div (A × B) = B. curl A − A. curl B curl (A × B) = A div B − B div A + (B.∇ )A − (A.∇ )B 23

¯

¯ c2 c3 ¯

grad (A.B) = A × curl B + B × curl A + (A.∇ )B + (B.∇ )A Integral Theorems Divergence theorem

Z

Stokes’ theorem

Z

surface

surface

A.dS =

Z

div A dV

volume

( curl A).dS =

I

contour

A.dr

Green’s theorems Z Z

volume

volume

n

(ψ∇

2

φ − φ∇

2

ψ)dV o

2

ψ∇ φ + (∇φ)(∇ψ) dV

=

Z

=

Z

surface

surface

!

Ã

∂ψ ∂φ −φ |dS| ψ ∂n ∂n

ψ

∂φ |dS| ∂n

where ˆ |dS| dS = n Green’s theorem in the plane I

(P dx + Qdy) =

Z Z Ã

24

∂Q ∂P − ∂x ∂y

!

dx dy

MECHANICS Kinematics Motion constant acceleration v = u + f t,

1 1 s = ut + f t2 = (u + v)t 2 2 v2 = u2 + 2f .s

General solution of

d2 x dt2

= −ω 2 x is x = a cos ωt + b sin ωt = R sin(ωt + φ)

where R =



a2 + b2 and cos φ = a/R, sin φ = b/R.

˙ = re ˙ θ and the acceleration is In polar coordinates the velocity is (r, ˙ r θ) ˙ r + rθe h i ˙ θ. r¨ − rθ˙2 , rθ¨ + 2r˙ θ˙ = (¨ r − r θ˙2 )er + (rθ¨ + 2r˙ θ)e Centres of mass The following results are for uniform bodies: 1 r 2 3 r 8 3 h 4

from vertex

arc, radius r and angle 2θ

(r sin θ)/θ

from centre

sector, radius r and angle 2θ

( 23 r sin θ)/θ

from centre

hemispherical shell, radius r hemisphere, radius r right circular cone, height h

from centre from centre

Moments of inertia i. The moment of inertia of a body of mass m about an axis = I + mh2 , where I is the moment of inertial about the parallel axis through the mass-centre and h is the distance between the axes. ii. If I1 and I2 are the moments of inertia of a lamina about two perpendicular axes through a point 0 in its plane, then its moment of inertia about the axis through 0 perpendicular to its plane is I1 + I2 . 25

iii. The following moments of inertia are for uniform bodies about the axes stated: rod, length `, through mid-point, perpendicular to rod

1 m`2 12 2

hoop, radius r, through centre, perpendicular to hoop

mr

disc, radius r, through centre, perpendicular to disc

1 mr2 2 2 mr2 5

sphere, radius r, diameter Work done W =

Z

tB tA

26

F.

dr dt. dt

ALGEBRAIC STRUCTURES

A group G is a set of elements {a, b, c, . . .} — with a binary operation ∗ such that i. a ∗ b is in G for all a, b in G ii. a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c in G iii. G contains an element e, called the identity element, such that e ∗ a = a = a ∗ e for all a in G

iv. given any a in G, there exists in G an element a−1 , called the element inverse to a, such that a−1 ∗ a = e = a ∗ a−1 . A commutative (or Abelian) group is one for which a ∗ b = b ∗ a for all a, b, in G. A field F is a set of elements {a, b, c, . . .} — with two binary operations + and . such that

i. F is a commutative group with respect to + with identity 0 ii. the non-zero elements of F form a commutative group with respect to . with identity 1 iii. a.(b + c) = a.b + a.c for all a, b, c, in F .

A vector space V over a field F is a set of elements {a, b, c, . . .} — with a binary

operation + such that

i. they form a commutative group under +; and, for all λ, µ in F and all a, b, in V , ii. λa is defined and is in V iii. λ(a + b) = λa + λb 27

iv. (λ + µ)a = λa + µa v. (λ.µ)a = λ(µa) vi. if 1 is an element of F such that 1.λ = λ for all λ in F , then 1a = a.

An equivalence relation R between the elements {a, b, c, . . .} — of a set C is a relation

such that, for all a, b, c in C

i. aRa (R is reflextive) ii. aRb ⇒ bRa (R is symmetric) iii. (aRb and bRc) ⇒ aRc (R is transitive).

28

PROBABILITY DISTRIBUTIONS

Name

Probability distribution /

Parameters

Mean

Variance

np

np(1 − p)

λ

λ

µ

σ2

1 λ

1 λ2

density function Binomial

n, p

³ ´ n r

P (X = r) =

pr (1 − p)n−r ,

r = 0, 1, 2, ..., n Poisson

λ

P (X = n) =

e−λ λn , n!

n = 0, 1, 2, ...... Normal

µ, σ

f (x) =

√1 σ 2π

exp{− 12

³

´ x−µ 2 }, σ

−∞ < x < ∞

f (x) = λe−λx ,

λ

Exponential

x > 0,

λ>0

THE F -DISTRIBUTION

The function tabulated on the next page is the inverse cumulative distribution function of Fisher’s F -distribution having ν1 and ν2 degrees of freedom. It is defined by P =

Γ Γ

³

³

1 ν 2 1

1 ν 2 1

´

+ 21 ν2 Γ

³

´

1 ν 2 2

1

´ ν12

ν1

1

ν22

ν2

Z

x 0

1

1

u 2 ν1 −1 (ν2 + ν1 u)− 2 (ν1 +ν2 ) du.

If X has an F -distribution with ν1 and ν2 degrees of freedom then P r.(X ≤ x) = P .

The table lists values of x for P = 0.95, P = 0.975 and P = 0.99, the upper number in each set being the value for P = 0.95.

29

ν2 1

2

3

4

5

6

7

8

9

10

12

15

20

25

50

100

ν1 : 1 ν 1 : 2 161 648 4052 18.51 38.51 98.50 10.13 17.44 34.12 7.71 12.22 21.20 6.61 10.01 16.26 5.99 8.81 13.75 5.59 8.07 12.25 5.32 7.57 11.26 5.12 7.21 10.56 4.96 6.94 10.04 4.75 6.55 9.33 4.54 6.20 8.68 4.35 5.87 8.10 4.24 5.69 7.77 4.03 5.34 7.17 3.94 5.18 6.90

199 799 5000 19.00 39.00 99.00 9.55 16.04 30.82 6.94 10.65 18.00 5.79 8.43 13.27 5.14 7.26 10.92 4.74 6.54 9.55 4.46 6.06 8.65 4.26 5.71 8.02 4.10 5.46 7.56 3.89 5.10 6.93 3.68 4.77 6.36 3.49 4.46 5.85 3.39 4.29 5.57 3.18 3.97 5.06 3.09 3.83 4.82

3

4

5

6

7

8

9

10

12

15

20

25

50

100

216 864 5403 19.16 39.17 99.17 9.28 15.44 29.46 6.59 9.98 16.69 5.41 7.76 12.06 4.76 6.60 9.78 4.35 5.89 8.45 4.07 5.42 7.59 3.86 5.08 6.99 3.71 4.83 6.55 3.49 4.47 5.95 3.29 4.15 5.42 3.10 3.86 4.94 2.99 3.69 4.68 2.79 3.39 4.20 2.70 3.25 3.98

225 900 5625 19.25 39.25 99.25 9.12 15.10 28.71 6.39 9.60 15.98 5.19 7.39 11.39 4.53 6.23 9.15 4.12 5.52 7.85 3.84 5.05 7.01 3.63 4.72 6.42 3.48 4.47 5.99 3.26 4.12 5.41 3.06 3.80 4.89 2.87 3.51 4.43 2.76 3.35 4.18 2.56 3.05 3.72 2.46 2.92 3.51

230 922 5764 19.30 39.30 99.30 9.01 14.88 28.24 6.26 9.36 15.52 5.05 7.15 10.97 4.39 5.99 8.75 3.97 5.29 7.46 3.69 4.82 6.63 3.48 4.48 6.06 3.33 4.24 5.64 3.11 3.89 5.06 2.90 3.58 4.56 2.71 3.29 4.10 2.60 3.13 3.85 2.40 2.83 3.41 2.31 2.70 3.21

234 937 5859 19.33 39.33 99.33 8.94 14.73 27.91 6.16 9.20 15.21 4.95 6.98 10.67 4.28 5.82 8.47 3.87 5.12 7.19 3.58 4.65 6.37 3.37 4.32 5.80 3.22 4.07 5.39 3.00 3.73 4.82 2.79 3.41 4.32 2.60 3.13 3.87 2.49 2.97 3.63 2.29 2.67 3.19 2.19 2.54 2.99

237 948 5928 19.35 39.36 99.36 8.89 14.62 27.67 6.09 9.07 14.98 4.88 6.85 10.46 4.21 5.70 8.26 3.79 4.99 6.99 3.50 4.53 6.18 3.29 4.20 5.61 3.14 3.95 5.20 2.91 3.61 4.64 2.71 3.29 4.14 2.51 3.01 3.70 2.40 2.85 3.46 2.20 2.55 3.02 2.10 2.42 2.82

239 957 5981 19.37 39.37 99.37 8.85 14.54 27.49 6.04 8.98 14.80 4.82 6.76 10.29 4.15 5.60 8.10 3.73 4.90 6.84 3.44 4.43 6.03 3.23 4.10 5.47 3.07 3.85 5.06 2.85 3.51 4.50 2.64 3.20 4.00 2.45 2.91 3.56 2.34 2.75 3.32 2.13 2.46 2.89 2.03 2.32 2.69

241 963 6022 19.38 39.39 99.39 8.81 14.47 27.35 6.00 8.90 14.66 4.77 6.68 10.16 4.10 5.52 7.98 3.68 4.82 6.72 3.39 4.36 5.91 3.18 4.03 5.35 3.02 3.78 4.94 2.80 3.44 4.39 2.59 3.12 3.89 2.39 2.84 3.46 2.28 2.68 3.22 2.07 2.38 2.78 1.97 2.24 2.59

242 969 6056 19.40 39.40 99.40 8.79 14.42 27.23 5.96 8.84 14.55 4.74 6.62 10.05 4.06 5.46 7.87 3.64 4.76 6.62 3.35 4.30 5.81 3.14 3.96 5.26 2.98 3.72 4.85 2.75 3.37 4.30 2.54 3.06 3.80 2.35 2.77 3.37 2.24 2.61 3.13 2.03 2.32 2.70 1.93 2.18 2.50

244 977 6106 19.41 39.41 99.42 8.74 14.34 27.05 5.91 8.75 14.37 4.68 6.52 9.89 4.00 5.37 7.72 3.57 4.67 6.47 3.28 4.20 5.67 3.07 3.87 5.11 2.91 3.62 4.71 2.69 3.28 4.16 2.48 2.96 3.67 2.28 2.68 3.23 2.16 2.51 2.99 1.95 2.22 2.56 1.85 2.08 2.37

246 985 6157 19.43 39.43 99.43 8.70 14.25 26.87 5.86 8.66 14.20 4.62 6.43 9.72 3.94 5.27 7.56 3.51 4.57 6.31 3.22 4.10 5.52 3.01 3.77 4.96 2.85 3.52 4.56 2.62 3.18 4.01 2.40 2.86 3.52 2.20 2.57 3.09 2.09 2.41 2.85 1.87 2.11 2.42 1.77 1.97 2.22

248 993 6209 19.45 39.45 99.45 8.66 14.17 26.69 5.80 8.56 14.02 4.56 6.33 9.55 3.87 5.17 7.40 3.44 4.47 6.16 3.15 4.00 5.36 2.94 3.67 4.81 2.77 3.42 4.41 2.54 3.07 3.86 2.33 2.76 3.37 2.12 2.46 2.94 2.01 2.30 2.70 1.78 1.99 2.27 1.68 1.85 2.07

249 998 6240 19.46 39.46 99.46 8.63 14.12 26.58 5.77 8.50 13.91 4.52 6.27 9.45 3.83 5.11 7.30 3.40 4.40 6.06 3.11 3.94 5.26 2.89 3.60 4.71 2.73 3.35 4.31 2.50 3.01 3.76 2.28 2.69 3.28 2.07 2.40 2.84 1.96 2.23 2.60 1.73 1.92 2.17 1.62 1.77 1.97

252 1008 6303 19.48 39.48 99.48 8.58 14.01 26.35 5.70 8.38 13.69 4.44 6.14 9.24 3.75 4.98 7.09 3.32 4.28 5.86 3.02 3.81 5.07 2.80 3.47 4.52 2.64 3.22 4.12 2.40 2.87 3.57 2.18 2.55 3.08 1.97 2.25 2.64 1.84 2.08 2.40 1.60 1.75 1.95 1.48 1.59 1.74

253 1013 6334 19.49 39.49 99.49 8.55 13.96 26.24 5.66 8.32 13.58 4.41 6.08 9.13 3.71 4.92 6.99 3.27 4.21 5.75 2.97 3.74 4.96 2.76 3.40 4.41 2.59 3.15 4.01 2.35 2.80 3.47 2.12 2.47 2.98 1.91 2.17 2.54 1.78 2.00 2.29 1.52 1.66 1.82 1.39 1.48 1.60

30

1

2

3

4

5

6

7

8

9

10

12

15

20

25

50

100

NORMAL DISTRIBUTION The function tabulated is the cumulative distribution function of a standard N (0, 1) random variable, namely

1 Z x − 1 t2 Φ(x) = √ e 2 dt. 2π −∞ If X is distributed N (0, 1) then Φ(x) = P r.(X ≤ x). x

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 0.1 0.2 0.3 0.4

0.5000 0.5398 0.5793 0.6179 0.6554

0.5040 0.5438 0.5832 0.6217 0.6591

0.5080 0.5478 0.5871 0.6255 0.6628

0.5120 0.5517 0.5910 0.6293 0.6664

0.5160 0.5557 0.5948 0.6331 0.6700

0.5199 0.5596 0.5987 0.6368 0.6736

0.5239 0.5636 0.6026 0.6406 0.6772

0.5279 0.5675 0.6064 0.6443 0.6808

0.5319 0.5714 0.6103 0.6480 0.6844

0.5359 0.5753 0.6141 0.6517 0.6879

0.5 0.6 0.7 0.8 0.9

0.6915 0.7257 0.7580 0.7881 0.8159

0.6950 0.7291 0.7611 0.7910 0.8186

0.6985 0.7324 0.7642 0.7939 0.8212

0.7019 0.7357 0.7673 0.7967 0.8238

0.7054 0.7389 0.7704 0.7995 0.8264

0.7088 0.7422 0.7734 0.8023 0.8289

0.7123 0.7454 0.7764 0.8051 0.8315

0.7157 0.7486 0.7794 0.8078 0.8340

0.7190 0.7517 0.7823 0.8106 0.8365

0.7224 0.7549 0.7852 0.8133 0.8389

1.0 1.1 1.2 1.3 1.4

0.8413 0.8643 0.8849 0.9032 0.9192

0.8438 0.8665 0.8869 0.9049 0.9207

0.8461 0.8686 0.8888 0.9066 0.9222

0.8485 0.8708 0.8907 0.9082 0.9236

0.8508 0.8729 0.8925 0.9099 0.9251

0.8531 0.8749 0.8944 0.9115 0.9265

0.8554 0.8770 0.8962 0.9131 0.9279

0.8577 0.8790 0.8980 0.9147 0.9292

0.8599 0.8810 0.8997 0.9162 0.9306

0.8621 0.8830 0.9015 0.9177 0.9319

1.5 1.6 1.7 1.8 1.9

0.9332 0.9452 0.9554 0.9641 0.9713

0.9345 0.9463 0.9564 0.9649 0.9719

0.9357 0.9474 0.9573 0.9656 0.9726

0.9370 0.9484 0.9582 0.9664 0.9732

0.9382 0.9495 0.9591 0.9671 0.9738

0.9394 0.9505 0.9599 0.9678 0.9744

0.9406 0.9515 0.9608 0.9686 0.9750

0.9418 0.9525 0.9616 0.9693 0.9756

0.9429 0.9535 0.9625 0.9699 0.9761

0.9441 0.9545 0.9633 0.9706 0.9767

2.0 2.1 2.2 2.3 2.4

0.9773 0.9821 0.9861 0.9893 0.9918

0.9778 0.9826 0.9864 0.9896 0.9920

0.9783 0.9830 0.9868 0.9898 0.9922

0.9788 0.9834 0.9871 0.9901 0.9925

0.9793 0.9838 0.9875 0.9904 0.9927

0.9798 0.9842 0.9878 0.9906 0.9929

0.9803 0.9846 0.9881 0.9909 0.9931

0.9808 0.9850 0.9884 0.9911 0.9932

0.9812 0.9854 0.9887 0.9913 0.9934

0.9817 0.9857 0.9890 0.9916 0.9936

2.5 2.6 2.7 2.8 2.9

0.9938 0.9953 0.9965 0.9974 0.9981

0.9940 0.9955 0.9966 0.9975 0.9982

0.9941 0.9956 0.9967 0.9976 0.9982

0.9943 0.9957 0.9968 0.9977 0.9983

0.9945 0.9959 0.9969 0.9977 0.9984

0.9946 0.9960 0.9970 0.9978 0.9984

0.9948 0.9961 0.9971 0.9979 0.9985

0.9949 0.9962 0.9972 0.9979 0.9985

0.9951 0.9963 0.9973 0.9980 0.9986

0.9952 0.9964 0.9974 0.9981 0.9986

3.0 3.1 3.2 3.3 3.4

0.9987 0.9990 0.9993 0.9995 0.9997

0.9987 0.9991 0.9993 0.9995 0.9997

0.9987 0.9991 0.9994 0.9995 0.9997

0.9988 0.9991 0.9994 0.9996 0.9997

0.9988 0.9992 0.9994 0.9996 0.9997

0.9989 0.9992 0.9994 0.9996 0.9997

0.9989 0.9992 0.9994 0.9996 0.9997

0.9989 0.9992 0.9995 0.9996 0.9997

0.9990 0.9993 0.9995 0.9996 0.9997

0.9990 0.9993 0.9995 0.9997 0.9998

3.5 3.6 3.7 3.8 3.9

0.9998 0.9998 0.9999 0.9999 1.0000

0.9998 0.9998 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

0.9998 0.9999 0.9999 0.9999 1.0000

31

THE t-DISTRIBUTION

The function tabulated is the inverse cumulative distribution function of Student’s t-distribution having ν degrees of freedom. It is defined by 1 1 Γ( 21 ν + 21 ) Z x P =√ (1 + t2 /ν)− 2 (ν+1) dt. 1 νπ Γ( 2 ν) −∞

If X has Student’s t-distribution with ν degrees of freedom then P r.(X ≤ x) = P . ν

P=0.90

P=0.95

0.975

0.990

0.995

0.999

0.9995

1 2 3 4 5

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

318.302 22.327 10.215 7.173 5.894

636.619 31.598 12.941 8.610 6.859

6 7 8 9 10

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

5.208 4.785 4.501 4.297 4.144

5.959 5.405 5.041 4.781 4.587

11 12 13 14 15

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

4.025 3.930 3.852 3.787 3.733

4.437 4.318 4.221 4.140 4.073

16 17 18 19 20

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

3.686 3.646 3.611 3.579 3.552

4.015 3.965 3.922 3.883 3.850

24 30 40 50 60

1.318 1.310 1.303 1.299 1.296

1.711 1.697 1.684 1.676 1.671

2.064 2.042 2.021 2.009 2.000

2.492 2.457 2.423 2.403 2.390

2.797 2.750 2.704 2.678 2.660

3.467 3.385 3.307 3.261 3.232

3.745 3.646 3.551 3.496 3.460

80 100 200 ∞

1.292 1.290 1.286 1.282

1.664 1.660 1.653 1.645

1.990 1.984 1.972 1.960

2.374 2.364 2.345 2.326

2.639 2.626 2.601 2.576

3.195 3.174 3.131 3.090

3.416 3.391 3.340 3.291

32

THE χ2 (CHI-SQUARED) DISTRIBUTION The function tabulated is the inverse cumulative distribution function of a Chisquared distribution having ν degrees of freedom. It is defined by Z x 1 1 1 ³ ´ P = u 2 ν−1 e− 2 u du. 1 ν/2 2 Γ 2ν 0

If X has an χ2 distribution with ν degrees of freedom then P r.(X ≤ x) = P . For √ √ ν > 100, 2X is approximately normally distributed with mean 2ν − 1 and unit variance. ν

P = 0.005

P = 0.01

0.025

0.05

0.950

0.975

0.990

0.995

0.999

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

0.04 393 0.010003 0.07172 0.2070 0.4117 0.6757 0.9893 1.344 1.735 2.156

0.03 157 0.02010 0.1148 0.2971 0.5543 0.8721 1.239 1.646 2.088 2.558

0.03 982 0.05064 0.2158 0.4844 0.8312 1.237 1.690 2.180 2.700 3.247

0.00393 0.1026 0.3518 0.7107 1.145 1.635 2.167 2.733 3.325 3.940

3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307

5.024 7.378 9.348 11.143 12.832 14.449 16.013 17.535 19.023 20.483

6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209

7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188

10.828 13.816 16.266 18.467 20.515 22.458 24.322 26.124 27.877 29.588

11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0

2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434

3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260

3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591

4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.117 10.851

19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410

21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170

24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566

26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997

31.264 32.909 34.528 36.123 37.697 39.252 40.790 42.312 43.820 45.315

21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0

8.034 8.643 9.260 9.886 10.520 11.160 11.808 12.461 13.121 13.787

8.897 9.542 10.196 10.856 11.524 12.198 12.879 13.565 14.256 14.953

10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791

11.591 12.338 13.091 13.848 14.611 15.379 16.151 16.928 17.708 18.493

32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773

35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979

38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892

41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672

46.797 48.268 49.728 51.179 52.620 54.052 55.476 56.892 58.301 59.703

40.0 50.0 60.0 70.0 80.0 90.0 100.0

20.707 27.991 35.534 43.275 51.172 59.196 67.328

22.164 29.707 37.485 45.442 53.540 61.754 70.065

24.433 32.357 40.482 48.758 57.153 65.647 74.222

26.509 34.764 43.188 51.739 60.391 69.126 77.929

55.758 67.505 79.082 90.531 101.879 113.145 124.342

59.342 71.420 83.298 95.023 106.629 118.136 129.561

63.691 76.154 88.379 100.425 112.329 124.116 135.807

66.766 79.490 91.952 104.215 116.321 128.299 140.169

73.402 86.661 99.607 112.317 124.839 137.208 149.449

33

PHYSICAL AND ASTRONOMICAL CONSTANTS 2.998 × 108 m s−1

c

Speed of light in vacuo

e

Elementary charge

mn

Neutron rest mass

mp

Proton rest mass

me

Electron rest mass

h

Planck’s constant

h ¯

Dirac’s constant (= h/2π)

k

Boltzmann’s constant

G

Gravitational constant

σ

Stefan-Boltzmann constant

c1

First Radiation Constant (= 2πhc2 )

c2

Second Radiation Constant (= hc/k) 1.439 × 10−2 m K

εo

1.602 × 10−19 C

1.675 × 10−27 kg

1.673 × 10−27 kg

9.110 × 10−31 kg

µo

Permeability of free scpae

NA

Avogadro constant

R

Gas constant

a0

Bohr radius

µB

Bohr magneton

α

Fine structure constant (= 1/137.0)



Solar Mass



Solar radius



Solar luminosity

M⊕

Earth Mass

R⊕

Mean earth radius

1 pc

Parsec

6.673 × 10−11 N m2 kg−2

5.670 × 10−8 J m−2 K−4 s−1 3.742 × 10−16 J m2 s−1

8.854 × 10−12 C2 N−1 m−2 4π × 10−7 H m−1

6.022 ×1023 mol−1

8.314 J K−1 mol−1 5.292 ×10−11 m

9.274 ×10−24 J T−1

7.297 ×10−3

1.989 ×1030 kg

6.96 ×108 m

3.827 ×1026 J s−1 5.976 ×1024 kg

6.371 ×106 m

9.461 ×1015 m

1 light year Astronomical Unit

1.055 × 10−34 J s

1.381 × 10−23 J K−1

Permittivity of free space

1 AU

6.626 × 10−34 J s

1.496 ×1011 m

3.086 ×1016 m 3.156 ×107 s

1 year

ENERGY CONVERSION : 1 joule (J) = 6.2415 × 1018 electronvolts (eV) 34

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University of Manchester MATHEMATICAL FORMULA TABLES

University of Manchester MATHEMATICAL FORMULA TABLES Version 2.1 November 2004 Other than this front cover these tables are identical to the UMIST, ...

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