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Mathematical Formula Handbook

Contents
Introduction
...
Series
...
Vector Algebra
...
Matrix Algebra
...
Vector Calculus
...
Complex Variables
...


6
...
10
Relations between sides and angles of any plane triangle;
Relations between sides and angles of any spherical triangle

7
...
11
Relations of the functions; Inverse functions

8
...
12
9
...
13
10
...
13
Standard forms; Standard substitutions; Integration by parts; Differentiation of an integral;
Dirac δ-‘function’; Reduction formulae

11
...
16
Diffusion (conduction) equation; Wave equation; Legendre’s equation; Bessel’s equation;
Laplace’s equation; Spherical harmonics

12
...
17
13
...
18
Taylor series for two variables; Stationary points; Changing variables: the chain rule;
Changing variables in surface and volume integrals – Jacobians

14
...
19
Fourier series; Fourier series for other ranges; Fourier series for odd and even functions;
Complex form of Fourier series; Discrete Fourier series; Fourier transforms; Convolution theorem;
Parseval’s theorem; Fourier transforms in two dimensions; Fourier transforms in three dimensions

15
...
23
16
...
24
Finding the zeros of equations; Numerical integration of differential equations;
Central difference notation; Approximating to derivatives; Interpolation: Everett’s formula;
Numerical evaluation of definite integrals

17
...
25
Range method; Combination of errors

18
...
26
Mean and Variance; Probability distributions; Weighted sums of random variables;
Statistics of a data sample x 1 ,
...
It is to some extent modelled on a similar
document issued by the Department of Engineering, but obviously reflects the particular interests of physicists
...
, but
a decision was taken against this, partly on the grounds that the book would become unduly bulky, but mainly
because, in its present form, clean copies can be made available to candidates in exams
...
Please send suggestions for amendments to
the Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition
...

This book was compiled by Dr John Shakeshaft and typeset originally by Fergus Gallagher, and currently by
Dr Dave Green, using the TEX typesetting package
...
5 December 2005
...
& Stegun, I
...
, Handbook of Mathematical Functions, Dover, 1965
...
S
...
M
...

Jahnke, E
...
, Tables of Functions, Dover, 1986
...
& Osterman, J
...

Speigel, M
...
, Mathematical Handbook of Formulas and Tables
...


Physical Constants
Based on the “Review of Particle Properties”, Barnett et al
...
(The figures in parentheses give the 1-standarddeviation uncertainties in the last digits
...
× 10 −12 F m−1

1·602 177 33(49) × 10 −19 C
6·626 075 5(40) × 10 −34 J s

1·054 572 66(63) × 10 −34 J s

6·022 136 7(36) × 10 23 mol−1

1·660 540 2(10) × 10 −27 kg

9·109 389 7(54) × 10 −31 kg

1·672 623 1(10) × 10 −27 kg

9·274 015 4(31) × 10 −24 J T−1

8·314 510(70) J K −1 mol−1

1·380 658(12) × 10 −23 J K−1

5·670 51(19) × 10 −8 W m−2 K−4
6·672 59(85) × 10 −11 N m2 kg−2

9·806 65 m s −2

(standard value at sea level)
1

1
...
P
...
P
...
)

1 − rn
,
1−r

n
[2a + (n − 1)d]
2
a
S∞ =
1−r

for |r| < 1

Convergence of series: the ratio test
Sn = u1 + u2 + u3 + · · · + un

converges as

n→∞

if

lim

n→∞

un+1
<1
un

Convergence of series: the comparison test
If each term in a series of positive terms is less than the corresponding term in a series known to be convergent,
then the given series is also convergent
...
When n is not a positive integer, the series does not terminate: the infinite series is
convergent for | x| < 1
...
A Maclaurin series is a Taylor series with
a = 0,
x2 d2 y
x3 d3 y
dy
+
+
+···
y( x) = y(0) + x
2
dx
2! dx
3! dx3

Power series with real variables
ex

ln(1 + x) =
cos x

=

sin x

=

tan x

=

tan−1 x

=

sin−1 x

2

x2
xn
+ ···+
+···
2!
n!
3
2
x
xn
x
+
+ · · · + (−1)n+1 + · · ·
x−
2
3
n
eix + e−ix
x2
x4
x6
= 1−
+
−
+ ···
2
2!
4!
6!
x3
x5
eix − e−ix
= x−
+
+ ···
2i
3!
5!
1
2 5
x + x3 +
x +···
3
15
x3
x5
x−
+
− ···
3
5
1
...
4 5

=1+x+

=

valid for all x
valid for −1 < x ≤ 1
valid for all values of x
valid for all values of x
valid for −

π
π
2
2

valid for −1 ≤ x ≤ 1
valid for −1 < x < 1

Integer series
N

∑n
1

= 1+ 2+ 3+ ···+ N =

N

N ( N + 1)
2

∑ n2 = 12 + 22 + 32 + · · · + N 2 =
1

N ( N + 1)(2N + 1)
6

N

∑ n3 = 13 + 23 + 33 + · · · + N 3 = [1 + 2 + 3 + · · · N ] 2 =
1

∞

∑
1

∞

∑
1

∞

N 2 ( N + 1)2
4

1
1
1
(−1)n+1
= 1 − + − + · · · = ln 2
n
2
3
4

[see expansion of ln (1 + x)]

1
1
1
π
(−1)n+1
= 1− + − + ··· =
2n − 1
3
5
7
4
1

∑ n2

=1+

1

[see expansion of tan−1 x]

1
1
1
π2
+ +
+··· =
4
9
16
6

N

∑ n(n + 1)(n + 2) = 1
...
3 + 2
...
4 + · · · + N ( N + 1)( N + 2) =
1

N ( N + 1)( N + 2)( N + 3)
4

This last result is a special case of the more general formula,
N

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

...


2
...
[Orthonormal vectors ≡
x
y
z
orthogonal unit vectors
...


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
...

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
...


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

Ay
By
Cy

Az
Bz = − A × C · B,
Cz

etc
...


e2 × e3
e1 · (e2 × e3 )

Summation convention
a

= ai ei

a·b

= ai bi

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

4

implies summation over i = 1
...
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
...
, ( I ) i j = δi j
...
Also I = I −1
...


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

( AB)i j =

∑ Aik Bk j

k=1

In general AB = BA
...


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
...


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

| A| =

∑

i jk
...


i, j,k,
...


2×2 matrices
If A =

a
c

b
d

then,

| A| = ad − bc

AT =

a
b

c
d

A−1 =

1
| A|

d −b
−c a

Product rules
( AB
...
B T A T
( AB
...
B−1 A−1
| AB
...
| N |

(if individual inverses exist)
(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
...


5

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
...
, 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)
...
If the columns of A are orthonormal vectors then
x = A T b
...
If A = A † then A is called a Hermitian matrix
...
The
eigenvalues are the zeros of the polynomial of degree n, Pn (λ ) = | A − λ I |
...
| A − λ I | = 0 is called the characteristic equation of the
matrix A
...

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
...


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∗ , b∗ ,
...
Vector Calculus
Notation
φ is a scalar function of a set of position coordinates
...

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
...

S = a surface area

τ = a volume contained by a specified 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)
Z

C

and when A =
Z

C

Z

A · t dL =

C

A · dL

φ
Z

( φ) · dL =

C

dφ

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

Z

also

(

( φ) dτ =

τ

τ

8

· A) dτ =
Z

Z

S

S

( A · n) dS =

φ dS

Z

S

A · dS
Z

τ

(

× A) dτ =

1
∂
∂θ
r sin θ

Z

S

(n × A) dS

sin θ

2

+

Transformation of integrals

Then

+

∂φ
∂θ

∂ 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

× A) · dS =

Z

C

φ) dS =

Z

C

(n ×

S

A · dL

φ dL

Green’s Theorem
Z

ψ φ · dS =

S

=

Z

Zτ

· (ψ φ) dτ
ψ

τ

2

φ + ( ψ) · ( φ) dτ

Green’s Second Theorem
Z

(ψ

2

τ

φ −φ

2

ψ) dτ =

Z

S

[ψ( φ) − φ( ψ)] · dS

5
...
The
real quantity r is the modulus of z and the angle θ is the argument of z
...

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
...

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
...
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 =

...

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
1
1
1
area = ab sin C = bc sin A = ca sin B =
2
2
2
cos A =

s(s − a)(s − b)(s − c),

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 b
sin c
sin a
=
=
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

10

1
( a + b + c)
2

7
...
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 ( a)
f ( x)
=
g( x)
g ( a)

(l’Hopital’s rule)
ˆ

9
...
Integration
Standard forms
Z

xn dx =

1
dx
x
Z
eax dx
Z

xn+1
+c
n+1

for n = −1
Z

1 ax
e +c
a
Z
x2
1
x ln x dx =
ln x −
2
2
Z
Z
Z
Z
Z

Z
Z
Z
Z

=

ln x dx = x(ln x − 1) + c

Z

= ln x + c

x eax dx = eax

x2 ± a2
x
x2 ± a2

dx = ln x +
dx =

a2 − x2 dx =

+c

+c

1
1
x
dx = tan−1
+c
a
a
a2 + x2
a+x
x
1
1
1
ln
dx = tanh−1
+c=
+c
a
a
2a
a−x
a2 − x2
1
1
1
x
x−a
dx = − coth −1
+c=
+c
ln
2
2
a
a
2a
x+a
x −a
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
1
x
dx = sin−1
+c
2 − x2
a
a
1

1
x
− 2
a
a

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

x2 ± a2 + c

x2 ± a2 + c
1
x
2

a2 − x2 + a2 sin −1

x
a

+c
13

Z
Z

∞
0
∞
0

1
dx = π cosec pπ
(1 + x) x p
cos( x2 ) dx =

Z

∞
0

for p < 1

sin ( x2 ) dx =

1
2

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

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

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

for n ≥ 1 and odd

Z

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

for n ≥ 2 and even

sin mx sin nx dx =

if m2 = n2
if m2 = n2

Standard substitutions
If the integrand is a function of:

( a2 − x2 ) or
( x2 + a2 ) or
( x2 − a2 ) or

a2 − x2

substitute:
x = a sin θ or x = a cos θ

x2 + a2

x = a tan θ or x = a sinh θ

x2 − a2

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

14

dx
( ax + b) px + q
dx

( ax + b)

px2

+ qx + r

dx =

substitute:
px + q = u2

ax + b =

1

...

1 + t2

Integration by parts
Z

b
a

b

Z

u dv = uv −
a

b
a

v du

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
...


Dirac δ-‘function’
1
δ (t − τ ) =
2π

Z

∞

−∞

exp[iω(t − τ )] dω
...


δ (t) dt = 1

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


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
...


(− 1/2)! =

√

π,

( 1/2)! =

√
π/ ,
2

etc
...

( 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

dθ =

π

...

2α

15

11
...

2

Recursion relation
Pl ( x) =

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

Orthogonality
Z

1

−1

Pl ( x) Pl ( x) dx =

2
δll
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
...


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

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;
d
d(cos θ )

Plm (cos θ ) = sin|m| θ

|m|

Pl (cos θ )

is the associated Legendre polynomial
...

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
...
e
...

8π

Orthogonality
Z

4π

Yl∗m Ylm dΩ = δll δmm

12
...


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

∂F
d
=
∂y
dx

dy
∂F
, where y =

...
Functions of Several Variables
∂φ
implies differentiation with respect to x keeping y, z,
...

∂x
∂φ
∂φ
∂φ
∂φ
∂φ
∂φ
dφ =
dx +
dy +
dz + · · · and δφ ≈
δx +
δy +
δz + · · ·
∂x
∂y
∂z
∂x
∂y
∂z

If φ = f ( x, y, z,
...
are independent variables
...

If φ is a well-behaved function then

∂φ
∂x

or
y,
...


∂2φ
∂ 2φ
=
etc
...


y

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
∂φ
1
+v
+
∂x
∂y
2!

u2

∂2φ
∂2φ
∂2φ
+ 2uv
+ v2 2
∂x ∂y
∂x2
∂y

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
...


∂2φ
∂2φ


> 0, or
> 0, 
Minimum:
2

∂2φ ∂2φ
∂2φ
∂x2
∂y2
>
and

∂x ∂y
∂2φ
∂2φ
∂x2 ∂y2

Maximum:
< 0, or
< 0, 

2
2
∂x
∂y
A function φ = f ( x, y) has a stationary point when

Saddle point:

If

∂2φ ∂2φ
<
∂x2 ∂y2

∂2φ
∂x ∂y

2

∂2φ
∂2φ
∂2φ
=
=
= 0 the character of the turning point is determined by the next higher derivative
...
) and the variables x, y,
...
then
∂φ
∂φ ∂x
∂φ ∂y
=
+
+ ···
∂u
∂x ∂u
∂y ∂u
∂φ ∂x
∂φ ∂y
∂φ
=
+
+ ···
∂v
∂x ∂v
∂y ∂v
etc
...
The corresponding formula for volume integrals is
∂(u, v)

f (u, v, w) J du dv dw

where now

∂x
∂u
∂y
J=
∂u
∂z
∂u

∂x
∂v
∂y
∂v
∂z
∂v

∂x
∂w
∂y
∂w
∂z
∂w

14
...
, M)
(m = 1,
...


Fourier series for other ranges
Variable t, range 0 ≤ t ≤ T, (i
...
, a periodic function of time with period T, frequency ω = 2π/ T)
...

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
...
e
...
If, in addition, y( x) is symmetric about
π 0
Z
4 π/ 2
y( x) sin mx dx (for m odd)
...
e
...
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
y( x) cos mx dx (for m odd)
...
]

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
...

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 x , range 0 ≤ x ≤ L,
∞

y( x ) =

∑ Cm e

i2mπx / L

,

−∞

Z

T

0

1
Cm =
L

y(t) e−imωt dt
...


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)
...


20

(m = 1,
...
, N − 1)

Fourier transforms
If y( x) is a function defined in the range −∞ ≤ x ≤ ∞ then the Fourier transform y(ω) is defined by the equations
Z ∞
Z ∞
1
iω t
y(ω) e dω,
y(t) e−iωt dt
...


If y(t) is symmetric about t = 0 then
Z ∞
Z
1 ∞
y(t) =
y(t) cos ωt dt
...

y(t) =
y(ω) = 2
π 0
0
Specific cases

y(t) = a,
= 0,

|t| ≤ τ
|t| > τ

(‘Top Hat’),

y(ω) = 2a

sin ωτ
≡ 2aτ sinc (ωτ )
ω
where

y(t) = a(1 − |t|/τ ),
= 0,

y(t) = exp(−t2 /t2 )
0

y(t) = f (t) eiω0 t

|t| ≤ τ
|t| > τ

(‘Saw-tooth’),

(Gaussian),

(modulated function),

y(ω) =

∑

m =− ∞

sin ( x)
x

2a
ωτ
(1 − cos ωτ ) = aτ sinc 2
2
2
ω τ

√
y(ω) = t0 π exp −ω2 t2 /4
0

y(ω) = f (ω − ω0 )

∞

y(t) =

sinc( x) =

∞

δ (t − mτ ) (sampling function)

y(ω) =

∑

n =− ∞

δ (ω − 2πn/τ )

21

Convolution theorem
If z(t) =

Z

∞

−∞

x(τ ) y(t − τ ) dτ =

Z

∞

−∞

x(t − τ ) y(τ ) dτ

≡ x(t) ∗ y(t) then

z (ω) = x(ω) y(ω)
...


Parseval’s theorem
Z

∞

−∞

y∗ (t) y(t) dt =

1
2π

Z

∞

−∞

y∗ (ω) y(ω) dω

(if y is normalised as on page 21)

Fourier transforms in two dimensions
V (k) =

Z

V (r ) e−ik·r d2 r

Z

∞

=

0

2πrV (r) J0 (kr) dr

if azimuthally symmetric

Fourier transforms in three dimensions
V (k) =

Z

V (r ) e−ik·r d3 r

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

if spherically symmetric

Examples
V (r )
1
4πr
e− λ r
4πr
V (r )
2

22

V (r )

V (k)
1
k2
1
2
k + λ2
ikV (k)

−k2 V (k)

15
...
]

23

16
...

xn+1 = xn −
f ( xn )
xn − xn−1
or, xn+1 = xn −
f ( xn )
f ( xn ) − f ( xn−1 )

(Newton)
(Linear interpolation)

are, in general, better approximations
...
The first two
terms represent linear interpolation
...


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

24

b

a

f ( x) dx ≈

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

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)
...


17
...


Range method
A quick but crude method of estimating σ is to find the range r of a set of n readings, i
...
, the difference between
the largest and smallest values, then
r
σ≈ √
...


Combination of errors
If Z = Z ( A, B,
...
independent) then

(σ Z )2 =

∂Z
σA
∂A

2

+

∂Z
σB
∂B

2

+···

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
σZ 2
σA 2
σB 2
=
+
Z
A
B
σZ
σA
=m
Z
A
σA
σZ =
A
σZ
= σA
Z

25

18
...
When the distribution of X is
discrete, the probability that X = x i is Pi
...

Mean µ = E( X ) = ∑ Pi xi or

Z

x 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
...
If X and Y are independent then V (W ) = a 2 V ( X ) + b2 V (Y )
...
, xn
Sample mean x =

1
n

∑ xi

Sample variance s 2 =

1
n

∑( 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) +
where the i are independent samples of a random variable with zero mean and variance σ 2
...


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


σ2
σ2
and 2

---

Title: mathmatics formula
Description: this is help to solve mathmatics problems

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