University of manchester mathematical formula tables

1. Department of Mathematics, UMIST
MATHEMATICAL FORMULA TABLES
Version 2.0 September 1999
1

2. 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
Ÿ
2
(Chi-squared) - Distribution 33
Physical and Astronomical constants 34
2

3. GREEK ALPHABET
A‹ alpha N — nu
BŒ beta Ξ ˜ xi
Í gamma O o omicron
∆Ždelta Π ™ pi
E; "epsilon P š rho
Z zeta Σ › sigma
H‘ eta T œ tau
Θ’; #theta Υ  upsilon
I“ iota Φ ž; 'phi
K”kappa X Ÿ chi
Λ• lambda Ψ   psi
M–mu Ω ! omega
INDICES AND LOGARITHMS
a
m
‚a
n
=a
m+n
(a
m
)
n
=a
mn
log(AB) = logA+ logB
log(A=B) = logA€logB
log(A
n
) =nlogA
log
ba=
log
ca
log
cb
3

4. TRIGONOMETRIC IDENTITIES
tanA= sinA=cosA
secA= 1=cosA
cosecA= 1=sinA
cotA= cosA=sinA= 1=tanA
sin
2
A+ cos
2
A= 1
sec
2
A= 1 + tan
2
A
cosec
2
A= 1 + cot
2
A
sin(A†B) = sinAcosB†cosAsinB
cos(A†B) = cosAcosB‡sinAsinB
tan(A†B) =
tanA†tanB
1‡tanAtanB
sin 2A= 2 sinAcosA
cos 2A= cos
2
A€sin
2
A
= 2 cos
2
A€1
= 1€2 sin
2
A
tan 2A=
2 tanA
1€tan
2
A
sin 3A= 3 sinA€4 sin
3
A
cos 3A= 4 cos
3
A€3 cosA
tan 3A=
3 tanA€tan
3
A
1€3 tan
2
A
sinA+ sinB= 2 sin
A+B
2
cos
A€B
2
4

5. sinA€sinB= 2 cos
A+B
2
sin
A€B
2
cosA+ cosB= 2 cos
A+B
2
cos
A€B
2
cosA€cosB=€2 sin
A+B
2
sin
A€B
2
2 sinAcosB= sin(A+B) + sin(A€B)
2 cosAsinB= sin(A+B)€sin(A€B)
2 cosAcosB= cos(A+B) + cos(A€B)
€2 sinAsinB= cos(A+B)€cos(A€B)
asinx+bcosx=Rsin(x+ž), whereR=
p
a
2
+b
2
and cosž=a=R, sinž=b=R.
Ift= tan
1
2
xthen sinx=
2t
1+t
2, cosx=
1€t
2
1+t
2.
cosx=
1
2
(e
ix
+e
€ix
) ; sinx=
1
2i
(e
ix
€e
€ix
)
e
ix
= cosx+isinx;e
€ix
= cosx€isinx
5

6. COMPLEX NUMBERS
i=
p
€1 Note:- ‘ j’ often used rather than ‘i’.
Exponential Notation
e
i’
= cos’+isin’
De Moivre’s theorem
[r(cos’+isin’)]
n
=r
n
(cosn’+isinn’)
n
th
roots of complex numbers
Ifz=re
i’
=r(cos’+isin’) then
z
1=n
=
n
p
re
i(’+2k™)=n
; k= 0;†1;†2; :::
HYPERBOLIC IDENTITIES
coshx= (e
x
+e
€x
)=2 sinh x= (e
x
€e
€x
)=2
tanhx= sinhx=coshx
sechx= 1=coshx cosechx= 1=sinhx
cothx= coshx=sinhx= 1=tanhx
coshix= cosx sinhix=isinx
cosix= coshx sinix=isinhx
cosh
2
A€sinh
2
A= 1
sech
2
A= 1€tanh
2
A
cosech
2
A= coth
2
A€1
6

7. SERIES
Powers of Natural Numbers
n
X
k=1
k=
1
2
n(n+ 1) ;
n
X
k=1
k
2
=
1
6
n(n+ 1)(2n+ 1);
n
X
k=1
k
3
=
1
4
n
2
(n+ 1)
2
Arithmetic Sn=
n€1
X
k=0
(a+kd) =
n
2
f2a+ (n€1)dg
Geometric (convergent for€1< r <1)
Sn=
n€1
X
k=0
ar
k
=
a(1€r
n
)
1€r
; S1=
a
1€r
Binomial (convergent forjxj<1)
(1 +x)
n
= 1 +nx+
n!
(n€2)!2!
x
2
+:::+
n!
(n€r)!r!
x
r
+:::
where
n!
(n€r)!r!
=
n(n€1)(n€2):::(n€r+ 1)
r!
Maclaurin series
f(x) =f(0) +xf
0
(0) +
x
2
2!
f
00
(0) +:::+
x
k
k!
f
(k)
(0) +Rk+1
whereRk+1=
x
k+1
(k+ 1)!
f
(k+1)
(’x);0< ’ <1
Taylor series
f(a+h) =f(a) +hf
0
(a) +
h
2
2!
f
00
(a) +:::+
h
k
k!
f
(k)
(a) +Rk+1
whereRk+1=
h
k+1
(k+ 1)!
f
(k+1)
(a+’h);0< ’ <1.
OR
f(x) =f(x
0
) + (x€x
0
)f
0
(x
0
) +
(x€x0)
2
2!
f
00
(x0) +:::+
(x€x0)
k
k!
f
(k)
(x0) +Rk+1
whereRk+1=
(x€x0)
k+1
(k+ 1)!
f
(k+1)
(x0+ (x€x0)’);0< ’ <1
7

8. Special Power Series
e
x
= 1 +x+
x
2
2!
+
x
3
3!
+:::+
x
r
r!
+::: (allx)
sinx=x€
x
3
3!
+
x
5
5!
€
x
7
7!
+:::+
(€1)
r
x
2r+1
(2r+ 1)!
+::: (allx)
cosx= 1€
x
2
2!
+
x
4
4!
€
x
6
6!
+:::+
(€1)
r
x
2r
(2r)!
+::: (allx)
tanx=x+
x
3
3
+
2x
5
15
+
17x
7
315
+::: (jxj<
™
2
)
sin
€1
x=x+
1
2
x
3
3
+
1:3
2:4
x
5
5
+
1:3:5
2:4:6
x
7
7
+
:::+
1:3:5::::(2n€1)
2:4:6::::(2n)
x
2n+1
2n+ 1
+::: (jxj<1)
tan
€1
x=x€
x
3
3
+
x
5
5
€
x
7
7
+:::+ (€1)
n
x
2n+1
2n+ 1
+:::(jxj<1)
`n(1 +x) =x€
x
2
2
+
x
3
3
€
x
4
4
+:::+ (€1)
n+1
x
n
n
+:::(€1< x”1)
sinhx=x+
x
3
3!
+
x
5
5!
+
x
7
7!
+:::+
x
2n+1
(2n+ 1)!
+::: (allx)
coshx= 1 +
x
2
2!
+
x
4
4!
+
x
6
6!
+:::+
x
2n
(2n)!
+::: (allx)
tanhx=x€
x
3
3
+
2x
5
15
€
17x
7
315
+::: (jxj<
™
2
)
sinh
€1
x=x€
1
2
x
3
3
+
1:3
2:4
x
5
5
€
1:3:5
2:4:6
x
7
7
+
:::+ (€1)
n
1:3:5:::(2n€1)
2:4:6:::2n
x
2n+1
2n+ 1
+:::(jxj<1)
tanh
€1
x=x+
x
3
3
+
x
5
5
+
x
7
7
+:::
x
2n+1
2n+ 1
+::: (jxj<1)
8

9. DERIVATIVES
function derivative
x
n
nx
n€1
e
x
e
x
a
x
(a >0) a
x
`na
`nx
1
x
log
ax
1
x`na
sinx cosx
cosx €sinx
tanx sec
2
x
cosecx €cosecxcotx
secx secxtanx
cotx €cosec
2
x
sin
€1
x
1
p
1€x
2
cos
€1
x €
1
p
1€x
2
tan
€1
x
1
1 +x
2
sinhx coshx
coshx sinhx
tanhx sech
2
x
cosechx €cosechxcothx
sechx €sechxtanhx
cothx €cosech
2
x
sinh
€1
x
1
p
1 +x
2
cosh
€1
x(x >1)
1
p
x
2
€1
tanh
€1
x(jxj<1)
1
1€x
2
coth
€1
x(jxj>1) €
1
x
2
€1
9

10. Product Rule
d
dx
(u(x)v(x)) =u(x)
dv
dx
+v(x)
du
dx
Quotient Rule
d
dx
 
u(x)
v(x)
!
=
v(x)
du
dx
€u(x)
dv
dx
[v(x)]
2
Chain Rule
d
dx
(f(g(x))) =f
0
(g(x))‚g
0
(x)
Leibnitz’s theorem
d
n
dx
n
(f:g) =f
(n)
:g+nf
(n€1)
:g
(1)
+
n(n€1)
2!
f
(n€2)
:g
(2)
+:::+
n!
(n€r)!r!
f
(n€r)
:g
(r)
+:::+f:g
(n)
10

11. INTEGRALS
function integral
f(x)
dg(x)
dx
f(x)g(x)€
Z
df(x)
dx
g(x)dx
x
n
(n6=€1)
x
n+1
n+1
1
x
`njxj Note:-`njxj+K=`njx=x0j
e
x
e
x
sinx €cosx
cosx sinx
tanx `n jsecxj
cosecx €`njcosecx+ cotxjor`n
Œ
Œ
Œtan
x
2
Œ
Œ
Œ
secx `n jsecx+ tanxj=`n
Œ
Œ
Œtan

™
4
+
x
2
‘Œ
Œ
Œ
cotx `n jsinxj
1
a
2
+x
2
1
a
tan
€1
x
a
1
a
2
€x
2
1
2a
`n
a+x
a€x
or
1
a
tanh
€1x
a
(jxj< a)
1
x
2
€a
2
1
2a
`n
x€a
x+a
or€
1
a
coth
€1x
a
(jxj> a)
1
p
a
2
€x
2
sin
€1
x
a
(a >jxj)
1
p
a
2
+x
2
sinh
€1x
a
or`n

x+
p
x
2
+a
2
‘
1
p
x
2
€a
2
cosh
€1x
a
or`njx+
p
x
2
€a
2
j (jxj> a)
sinhx coshx
coshx sinhx
tanhx `n coshx
cosechx €`njcosechx+cothxjor`n
Œ
Œ
Œtanh
x
2
Œ
Œ
Œ
sechx 2 tan
€1
e
x
cothx `n jsinhxj
11

12. Double integral
Z Z
f(x; y)dxdy=
Z Z
g(r; s)Jdrds
where
J=
@(x; y)
@(r; s)
=
Œ
Œ
Œ
Œ
Œ
Œ
Œ
@x
@r
@x
@s
@y
@r
@y
@s
Œ
Œ
Œ
Œ
Œ
Œ
Œ
12

13. LAPLACE TRANSFORMS
˜
f(s) =
R
1
0
e
€st
f(t)dt
function transform
1
1
s
t
n
n!
s
n+1
e
at
1
s€a
sin!t
!
s
2
+!
2
cos!t
s
s
2
+!
2
sinh!t
!
s
2
€!
2
cosh!t
s
s
2
€!
2
tsin!t
2!s
(s
2
+!
2
)
2
tcos!t
s
2
€!
2
(s
2
+!
2
)
2
Ha(t) =H(t€a)
e
€as
s
Ž(t) 1
e
at
t
n
n!
(s€a)
n+1
e
at
sin!t
!
(s€a)
2
+!
2
e
at
cos!t
s€a
(s€a)
2
+!
2
e
at
sinh!t
!
(s€a)
2
€!
2
e
at
cosh!t
s€a
(s€a)
2
€!
2
13

14. Let
˜
f(s) =L ff(t)gthen
L
n
e
at
f(t)
o
=
˜
f(s€a);
L ftf(t)g=€
d
ds
(
˜
f(s));
L
(
f(t)
t
)
=
Z
1
x=s
˜
f(x)dxif this exists:
Derivatives and integrals
Lety=y(t) and let ˜y=L fy(t)gthen
L
(
dy
dt
)
=s˜y€y0;
L
(
d
2
y
dt
2
)
=s
2
˜y€sy0€y
0
0;
L
šZ
t
œ=0
y(œ)dœ
›
=
1
s
˜y
wherey0andy
0
0are the values ofyanddy=dtrespectively att= 0.
Time delay
Let g(t) =Ha(t)f(t€a) =
8
>
<
>
:
0 t < a
f(t€a)t > a
then L fg(t)g=e
€as˜
f(s):
Scale change
L ff(kt)g=
1
k
˜
f
’
s
k
“
:
Periodic functions
Letf(t) be of periodTthen
L ff(t)g=
1
1€e
€sT
Z
T
t=0
e
€st
f(t)dt:
14

15. Convolution
Letf(t)ƒg(t) =
R
t
x=0
f(x)g(t€x)dx=
R
t
x=0
f(t€x)g(x)dx
then L ff(t)ƒg(t)g=
˜
f(s)˜g(s).
RLC circuit
For a simple RLC circuit with initial chargeq0and initial currenti0,
˜
E=
’
r+Ls+
1
Cs
“
e
i€Li0+
1
Cs
q0:
Limiting values
initial value theorem
lim
t!0+
f(t) = lim
s!1
s
˜
f(s);
final value theorem
lim
t!1
f(t) = lim
s!0+
s
˜
f(s);
Z
1
0
f(t)dt= lim
s!0+
˜
f(s)
provided these limits exist.
15

16. ZTRANSFORMS
Zff(t)g=
˜
f(z) =
1
X
k=0
f(kT)z
€k
function transform
Žt;nT z
€n
(n•0)
e
€at
z
z€e
€aT
te
€at
Tze
€aT
(z€e
€aT
)
2
t
2
e
€at
T
2
ze
€aT
(z+e
€aT
)
(z€e
€aT
)
3
sinhat
zsinhaT
z
2
€2zcoshaT+ 1
coshat
z(z€coshaT)
z
2
€2zcoshaT+ 1
e
€at
sin!t
ze
€aT
sin!T
z
2
€2ze
€aT
cos!T+e
€2aT
e
€at
cos!t
z(z€e
€aT
cos!T)
z
2
€2ze
€aT
cos!T+e
€2aT
te
€at
sin!t
Tze
€aT
(z
2
€e
€2aT
) sin!T
(z
2
€2ze
€aT
cos!T+e
€2aT
)
2
te
€at
cos!t
Tze
€aT
(z
2
cos!T€2ze
€aT
+e
€2aT
cos!T)
(z
2
€2ze
€aT
cos!T+e
€2aT
)
2
Shift Theorem
Zff(t+nT)g=z
n˜
f(z)€
P
n€1
k=0
z
n€k
f(kT) (n >0)
Initial value theorem
f(0) = limz!1
˜
f(z)
16

17. Final value theorem
f(1) = lim
z!1
h
(z€1)
˜
f(z)
i
provided f(1)exists:
Inverse Formula
f(kT) =
1
2™
Z
™
€™
e
ik’˜
f(e
i’
)d’
FOURIER SERIES AND TRANSFORMS
Fourier series
f(t) =
1
2
a0+
1
X
n=1
fancosn!t+bnsinn!tg (periodT= 2™=!)
where
an=
2
T
Z
t0+T
t0
f(t) cosn!t dt
bn=
2
T
Z
t0+T
t0
f(t) sinn!t dt
17

18. Half range Fourier series
sine series a n= 0; bn=
4
T
Z
T =2
0
f(t) sinn!t dt
cosine series b n= 0; an=
4
T
Z
T =2
0
f(t) cosn!t dt
Finite Fourier transforms
sine transform
˜
fs(n) =
4
T
Z
T =2
0
f(t) sinn!t dt
f(t) =
1
X
n=1
˜
fs(n) sinn!t
cosine transform
˜
fc(n) =
4
T
Z
T =2
0
f(t) cosn!t dt
f(t) =
1
2
˜
fc(0) +
1
X
n=1
˜
fc(n) cosn!t
Fourier integral
1
2
’
lim
t%0
f(t) + lim
t&0
f(t)
“
=
1
2™
Z
1
€1
e
i!t
Z
1
€1
f(u)e
€i!u
du d!
Fourier integral transform
˜
f(!) =Fff(t)g=
1
p
2™
Z
1
€1
e
€i!u
f(u)du
f(t) =F
€1
n
˜
f(!)
o
=
1
p
2™
Z
1
€1
e
i!t˜
f(!)d!
18

19. NUMERICAL FORMULAE
Iteration
Newton Raphson method for refining an approximate rootx0off(x) = 0
xn+1=xn€
f(xn)
f
0
(xn)
Particular case to find
p
Nusexn+1=
1
2

xn+
N
xn
‘
.
Secant Method
xn+1=xn€f(xn)=
 
f(xn)€f(xn€1)
xn€xn€1
!
Interpolation
∆fn=fn+1€fn; Žfn=f
n+
1
2
€f
n€
1
2
rfn=fn€fn€1; –fn=
1
2

f
n+
1
2
+f
n€
1
2
‘
Gregory Newton Formula
fp=f0+p∆f0+
p(p€1)
2!
∆
2
f0+:::+
p!
(p€r)!r!
∆
r
f0
wherep=
x€x0
h
Lagrange’s Formulafornpoints
y=
n
X
i=1
yi`i(x)
where
`i(x) =
Π
n
j=1;j6=i(x€xj)
Π
n
j=1;j6=i
(xi€xj)
19

20. Numerical differentiation
Derivatives at a tabular point
hf
0
0=–Žf0€
1
6
–Ž
3
f0+
1
30
–Ž
5
f0€:::
h
2
f
00
0=Ž
2
f0€
1
12
Ž
4
f0+
1
90
Ž
6
f0€:::
hf
0
0= ∆f0€
1
2
∆
2
f0+
1
3
∆
3
f0€
1
4
∆
4
f0+
1
5
∆
5
f0€:::
h
2
f
00
0= ∆
2
f0€∆
3
f0+
11
12
∆
4
f0€
5
6
∆
5
f0+:::
Numerical Integration
TrapeziumRule
Z
x0+h
x0
f(x)dx'
h
2
(f0+f1) +E
where f i=f(x0+ih); E=€
h
3
12
f
00
(a); x0< a < x0+h
Composite Trapezium Rule
Z
x0+nh
x0
f(x)dx'
h
2
ff0+ 2f1+ 2f2+:::2fn€1+fng €
h
2
12
(f
0
n€f
0
0) +
h
4
720
(f
000
n€f
000
0):::
wheref
0
0=f
0
(x0); f
0
n=f
0
(x0+nh), etc
Simpson
0
sRule
Z
x0+2h
x0
f(x)dx'
h
3
(f0+4f1+f2)+E
where E =€
h
5
90
f
(4)
(a) x0< a < x0+ 2h:
Composite Simpson’s Rule(neven)
Z
x0+nh
x0
f(x)dx'
h
3
(f0+ 4f1+ 2f2+ 4f3+ 2f4+:::+ 2fn€2+ 4fn€1+fn) +E
where E =€
nh
5
180
f
(4)
(a): x0< a < x0+nh
20

21. Gauss order 1.(Midpoint)
Z
1
€1
f(x)dx= 2f(0) +E
where E =
2
3
f
00
(a):€1< a <1
Gauss order 2.
Z
1
€1
f(x)dx=f
 
€
1
p
3
!
+f
 
1
p
3
!
+E
where E =
1
135
f
0v
(a):€1< a <1
Differential Equations
To solvey
0
=f(x; y) given initial conditiony0atx0; 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)
yn+1=yn+
h
2
(f(xn; yn) +f(xn+h; yn+hf(xn; yn))):
Runge Kutta order 4.
yn+1=yn+
h
6
(K1+ 2K2+ 2K3+K4)
where
K1=f(xn; yn)
K2=f
 
xn+
h
2
; yn+
hK1
2
!
K3=f
 
xn+
h
2
; yn+
hK2
2
!
K4=f(xn+h; yn+hK3)
21

22. Chebyshev Polynomials
Tn(x) = cos
h
n(cos
€1
x)
i
To(x) = 1 T1(x) =x
Un€1(x) =
T
0
n(x)
n
=
sin [n(cos
€1
x)]
p
1€x
2
Tm(Tn(x)) =Tmn(x):
Tn+1(x) = 2xTn(x)€Tn€1(x)
Un+1(x) = 2xUn(x)€Un€1(x)
Z
Tn(x)dx=
1
2
(
Tn+1(x)
n+ 1
€
Tn€1(x)
n€1
)
+constant; n•2
f(x) =
1
2
a0T0(x) +a1T1(x):::ajTj(x) +:::
where a j=
2
™
Z
™
0
f(cos’) cosj’d’ j •0
and
R
f(x)dx= constant +A1T1(x) +A2T2(x) +:::AjTj(x) +:::
whereAj= (aj€1€aj+1)=2j j •1
22

23. VECTOR FORMULAE
Scalar producta:b=abcos’=a1b1+a2b2+a3b3
Vector producta‚b=absin’ˆn=
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
i j k
a1a2a3
b1b2b3
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
= (a2b3€a3b2)i+ (a3b1€a1b3)j+ (a1b2€a2b1)k
Triple products
[a;b;c] = (a‚b):c=a:(b‚c) =
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
a1a2a3
b1b2b3
c1c2c3
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
Œ
a‚(b‚c) = (a:c)b€(a:b)c
Vector Calculus
r ‘
 
@
@x
;
@
@y
;
@
@z
!
gradž‘ rž;divA‘ r:A;curlA‘ r ‚A
div gradž‘ r:(rž)‘ r
2
ž(for scalars only)
div curlA= 0 curl grad ž‘0
r
2
A= grad divA€curl curlA
r(‹Œ) =‹rŒ+Œr‹
div (‹A) =‹divA+A:(r‹)
curl (‹A) =‹curlA€A‚(r‹)
div (A‚B) =B:curlA€A:curlB
curl (A‚B) =AdivB€BdivA+ (B:r)A€(A:r)B
23

24. grad (A:B) =A‚curlB+B‚curlA+ (A:r)B+ (B:r)A
Integral Theorems
Divergence theorem
Z
surface
A:dS=
Z
volume
divAdV
Stokes’ theorem
Z
surface
( curlA):dS=
I
contour
A:dr
Green’s theorems
Z
volume
( r
2
ž€žr
2
 )dV=
Z
surface
 
 
@ž
@n
€ž
@ 
@n
!
jdSj
Z
volume
n
 r
2
ž+ (rž)(r )
o
dV=
Z
surface
 
@ž
@n
jdSj
where
dS=ˆnjdSj
Green’s theorem in the plane
I
(Pdx+Qdy) =
Z Z
 
@Q
@x
€
@P
@y
!
dxdy
24

25. MECHANICS
Kinematics
Motion constant acceleration
v=u+ft;s=ut+
1
2
ft
2
=
1
2
(u+v)t
v
2
=u
2
+ 2f:s
General solution of
d
2
x
dt
2=€!
2
xis
x=acos!t+bsin!t=Rsin(!t+ž)
whereR=
p
a
2
+b
2
and cosž=a=R, sinž=b=R.
In polar coordinates the velocity is ( ˙r; r
˙
’) = ˙rer+r
˙
’e’and the acceleration is
h
¨r€r
˙
’
2
; r
¨
’+ 2 ˙r
˙
’
i
= (¨r€r
˙
’
2
)er+ (r
¨
’+ 2 ˙r
˙
’)e’.
Centres of mass
The following results are for uniform bodies:
hemispherical shell, radiusr
1
2
r from centre
hemisphere, radiusr
3
8
r from centre
right circular cone, heighth
3
4
h from vertex
arc, radiusrand angle 2’ (rsin’)=’from centre
sector, radiusrand angle 2’(
2
3
rsin’)=’from centre
Moments of inertia
i. The moment of inertia of a body of massmabout an axis =I+mh
2
, whereI
is the moment of inertial about the parallel axis through the mass-centre andh
is the distance between the axes.
ii. IfI1andI2are the moments of inertia ofalamina 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 isI1+I2.
25

26. iii. The following moments of inertia are for uniform bodies about the axes stated:
rod, length`, through mid-point, perpendicular to rod
1
12
m`
2
hoop, radiusr, through centre, perpendicular to hoopmr
2
disc, radiusr, through centre, perpendicular to disc
1
2
mr
2
sphere, radiusr, diameter
2
5
mr
2
Work done
W=
Z
tB
tA
F:
dr
dt
dt:
26

27. ALGEBRAIC STRUCTURES
A groupGis a set of elementsfa; b; c; : : :g— with a binary operationƒsuch that
i.aƒbis inGfor alla; binG
ii.aƒ(bƒc) = (aƒb)ƒcfor alla; b; cinG
iii.Gcontains an elemente, called the identity element, such thateƒa=a=aƒe
for allainG
iv. given anyainG, there exists inGan elementa
€1
, called the element inverse
toa, such thata
€1
ƒa=e=aƒa
€1
.
A commutative(or Abelian) groupis one for whichaƒb=bƒafor alla; b, inG.
A fieldFis a set of elementsfa; b; c; : : :g— with two binary operations + and . such
that
i.Fis a commutative group with respect to + with identity 0
ii. the non-zero elements ofFform a commutative group with respect to . with
identity 1
iii.a:(b+c) =a:b+a:cfor alla; b; c, inF.
A vector spaceVover a fieldFis a set of elementsfa; b; c; : : :g— with a binary
operation + such that
i. they form a commutative group under +;
and, for all•; –inFand alla; b, inV,
ii.•ais defined and is inV
iii.•(a+b) =•a+•b
27

28. iv. (•+–)a=•a+–a
v. (•:–)a=•(–a)
vi. if 1 is an element ofFsuch that 1:•=•for all•inF, then 1a=a.
An equivalence relationRbetween the elementsfa; b; c; : : :g— of a setCis a relation
such that, for alla; b; cinC
i.aRa(Ris reflextive)
ii.aRb)bRa(Ris symmetric)
iii. (aRbandbRc))aRc(Ris transitive).
28

29. PROBABILITY DISTRIBUTIONS
Name Parameters Probability distribution /MeanVariance
density function
Binomial n; p P(X=r) =
n!
(n€r)!r!
p
r
(1€p)
n€r
;npnp(1€p)
r= 0;1;2; :::; n
Poisson • P(X=n) =
e
ۥ
•
n
n!
; • •
n= 0;1;2; ::::::
Normal –; › f(x) =
1
›
p
2™
expf€
1
2

x€–
›
‘
2
g; – ›
2
€1< x <1
Exponential • f(x) =•e
ۥx
;
1
•
1
•
2
x >0; • >0
THEF-DISTRIBUTION
The function tabulated on the next page is the inverse cumulative distribution
function of Fisher’sF-distribution having—1and—2degrees of freedom. It is defined
by
P=
Γ

1
2
—1+
1
2
—2
‘
Γ

1
2
—1
‘
Γ

1
2
—2
‘—
1
2
—1
1—
1
2
—2
2
Z
x
0
u
1
2
—1€1
(—2+—1u)
€
1
2
(—1+—2)
du:
IfXhas anF-distribution with—1and—2degrees of freedom thenPr:(X”x) =P.
The table lists values ofxforP= 0:95,P= 0:975 andP= 0:99, the upper number
in each set being the value forP= 0:95.
29

30. —2—1: 1—1: 23456789101215202550100
161199216225230234237239241242244246248249252253
1648799864900922937948957963969977985993998100810131
4052500054035625576458595928598160226056610661576209624063036334
18.5119.0019.1619.2519.3019.3319.3519.3719.3819.4019.4119.4319.4519.4619.4819.49
238.5139.0039.1739.2539.3039.3339.3639.3739.3939.4039.4139.4339.4539.4639.4839.492
98.5099.0099.1799.2599.3099.3399.3699.3799.3999.4099.4299.4399.4599.4699.4899.49
10.139.559.289.129.018.948.898.858.818.798.748.708.668.638.588.55
317.4416.0415.4415.1014.8814.7314.6214.5414.4714.4214.3414.2514.1714.1214.0113.963
34.1230.8229.4628.7128.2427.9127.6727.4927.3527.2327.0526.8726.6926.5826.3526.24
7.716.946.596.396.266.166.096.046.005.965.915.865.805.775.705.66
412.2210.659.989.609.369.209.078.988.908.848.758.668.568.508.388.324
21.2018.0016.6915.9815.5215.2114.9814.8014.6614.5514.3714.2014.0213.9113.6913.58
6.615.795.415.195.054.954.884.824.774.744.684.624.564.524.444.41
510.018.437.767.397.156.986.856.766.686.626.526.436.336.276.146.085
16.2613.2712.0611.3910.9710.6710.4610.2910.1610.059.899.729.559.459.249.13
5.995.144.764.534.394.284.214.154.104.064.003.943.873.833.753.71
68.817.266.606.235.995.825.705.605.525.465.375.275.175.114.984.926
13.7510.929.789.158.758.478.268.107.987.877.727.567.407.307.096.99
5.594.744.354.123.973.873.793.733.683.643.573.513.443.403.323.27
78.076.545.895.525.295.124.994.904.824.764.674.574.474.404.284.217
12.259.558.457.857.467.196.996.846.726.626.476.316.166.065.865.75
5.324.464.073.843.693.583.503.443.393.353.283.223.153.113.022.97
87.576.065.425.054.824.654.534.434.364.304.204.104.003.943.813.748
11.268.657.597.016.636.376.186.035.915.815.675.525.365.265.074.96
5.124.263.863.633.483.373.293.233.183.143.073.012.942.892.802.76
97.215.715.084.724.484.324.204.104.033.963.873.773.673.603.473.409
10.568.026.996.426.065.805.615.475.355.265.114.964.814.714.524.41
4.964.103.713.483.333.223.143.073.022.982.912.852.772.732.642.59
106.945.464.834.474.244.073.953.853.783.723.623.523.423.353.223.1510
10.047.566.555.995.645.395.205.064.944.854.714.564.414.314.124.01
4.753.893.493.263.113.002.912.852.802.752.692.622.542.502.402.35
126.555.104.474.123.893.733.613.513.443.373.283.183.073.012.872.8012
9.336.935.955.415.064.824.644.504.394.304.164.013.863.763.573.47
4.543.683.293.062.902.792.712.642.592.542.482.402.332.282.182.12
156.204.774.153.803.583.413.293.203.123.062.962.862.762.692.552.4715
8.686.365.424.894.564.324.144.003.893.803.673.523.373.283.082.98
4.353.493.102.872.712.602.512.452.392.352.282.202.122.071.971.91
205.874.463.863.513.293.133.012.912.842.772.682.572.462.402.252.1720
8.105.854.944.434.103.873.703.563.463.373.233.092.942.842.642.54
4.243.392.992.762.602.492.402.342.282.242.162.092.011.961.841.78
255.694.293.693.353.132.972.852.752.682.612.512.412.302.232.082.0025
7.775.574.684.183.853.633.463.323.223.132.992.852.702.602.402.29
4.033.182.792.562.402.292.202.132.072.031.951.871.781.731.601.52
505.343.973.393.052.832.672.552.462.382.322.222.111.991.921.751.6650
7.175.064.203.723.413.193.022.892.782.702.562.422.272.171.951.82
3.943.092.702.462.312.192.102.031.971.931.851.771.681.621.481.39
1005.183.833.252.922.702.542.422.322.242.182.081.971.851.771.591.48100
6.904.823.983.513.212.992.822.692.592.502.372.222.071.971.741.60
30

31. NORMAL DISTRIBUTION
The function tabulated is the cumulative distribution function of a standardN(0;1)
random variable, namely
Φ(x) =
1
p
2™
Z
x
€1
e
€
1
2
t
2
dt:
IfXis distributedN(0;1) then Φ(x) =Pr:(X”x).
x0.000.010.020.030.040.050.060.070.080.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
2.00.97730.97780.97830.97880.97930.97980.98030.98080.98120.9817
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
3.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993
3.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995
3.30.99950.99950.99950.99960.99960.99960.99960.99960.99960.9997
3.40.99970.99970.99970.99970.99970.99970.99970.99970.99970.9998
3.50.99980.99980.99980.99980.99980.99980.99980.99980.99980.9998
3.60.99980.99980.99990.99990.99990.99990.99990.99990.99990.9999
3.70.99990.99990.99990.99990.99990.99990.99990.99990.99990.9999
3.80.99990.99990.99990.99990.99990.99990.99990.99990.99990.9999
3.91.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
31

32. THEt-DISTRIBUTION
The function tabulated is the inverse cumulative distribution function of Student’s
t-distribution having—degrees of freedom. It is defined by
P=
1
p
—™
Γ(
1
2
—+
1
2
)
Γ(
1
2
—)
Z
x
€1
(1 +t
2
=—)
€
1
2
(—+1)
dt:
IfXhas Student’st-distribution with—degrees of freedom thenPr:(X”x) =P.
—P=0.90P=0.950.9750.9900.9950.9990.9995
1 3.078 6.314 12.70631.82163.657318.302636.619
2 1.886 2.920 4.3036.9659.92522.32731.598
3 1.638 2.353 3.1824.5415.84110.21512.941
4 1.533 2.132 2.7763.7474.6047.1738.610
5 1.476 2.015 2.5713.3654.0325.8946.859
6 1.440 1.943 2.4473.1433.7075.2085.959
7 1.415 1.895 2.3652.9983.4994.7855.405
8 1.397 1.860 2.3062.8963.3554.5015.041
9 1.383 1.833 2.2622.8213.2504.2974.781
101.372 1.812 2.2282.7643.1694.1444.587
111.363 1.796 2.2012.7183.1064.0254.437
121.356 1.782 2.1792.6813.0553.9304.318
131.350 1.771 2.1602.6503.0123.8524.221
141.345 1.761 2.1452.6242.9773.7874.140
151.341 1.753 2.1312.6022.9473.7334.073
161.337 1.746 2.1202.5832.9213.6864.015
171.333 1.740 2.1102.5672.8983.6463.965
181.330 1.734 2.1012.5522.8783.6113.922
191.328 1.729 2.0932.5392.8613.5793.883
201.325 1.725 2.0862.5282.8453.5523.850
241.318 1.711 2.0642.4922.7973.4673.745
301.310 1.697 2.0422.4572.7503.3853.646
401.303 1.684 2.0212.4232.7043.3073.551
501.299 1.676 2.0092.4032.6783.2613.496
601.296 1.671 2.0002.3902.6603.2323.460
801.292 1.664 1.9902.3742.6393.1953.416
1001.290 1.660 1.9842.3642.6263.1743.391
2001.286 1.653 1.9722.3452.6013.1313.340
1 1.282 1.645 1.9602.3262.5763.0903.291
32

33. THEŸ
2
(CHI-SQUARED) DISTRIBUTION
The function tabulated is the inverse cumulative distribution function of a Chi-
squared distribution having—degrees of freedom. It is defined by
P=
1
2
—=2
Γ

1
2
—
‘
Z
x
0
u
1
2
—€1
e
€
1
2
u
du:
IfXhas anŸ
2
distribution with—degrees of freedom thenPr:(X”x) =P. For
— >100,
p
2Xis approximately normally distributed with mean
p
2—€1 and unit
variance.
— P = 0.005P = 0.010.0250.05 0.9500.9750.9900.9950.999
1.00:0
4
393 0:0
3
1570:0
3
9820.003933.8415.0246.6357.87910.828
2.00.0100030.020100.050640.10265.9917.3789.21010.59713.816
3.00.07172 0.11480.21580.35187.8159.34811.34512.83816.266
4.00.2070 0.29710.48440.71079.48811.14313.27714.86018.467
5.00.4117 0.55430.83121.14511.07012.83215.08616.75020.515
6.00.6757 0.87211.2371.63512.59214.44916.81218.54822.458
7.00.9893 1.239 1.6902.16714.06716.01318.47520.27824.322
8.01.344 1.646 2.1802.73315.50717.53520.09021.95526.124
9.01.735 2.088 2.7003.32516.91919.02321.66623.58927.877
10.02.156 2.558 3.2473.94018.30720.48323.20925.18829.588
11.02.603 3.053 3.8164.57519.67521.92024.72526.75731.264
12.03.074 3.571 4.4045.22621.02623.33726.21728.30032.909
13.03.565 4.107 5.0095.89222.36224.73627.68829.81934.528
14.04.075 4.660 5.6296.57123.68526.11929.14131.31936.123
15.04.601 5.229 6.2627.26124.99627.48830.57832.80137.697
16.05.142 5.812 6.9087.96226.29628.84532.00034.26739.252
17.05.697 6.408 7.5648.67227.58730.19133.40935.71840.790
18.06.265 7.015 8.2319.39028.86931.52634.80537.15642.312
19.06.844 7.633 8.90710.11730.14432.85236.19138.58243.820
20.07.434 8.260 9.59110.85131.41034.17037.56639.99745.315
21.08.034 8.897 10.28311.59132.67135.47938.93241.40146.797
22.08.643 9.542 10.98212.33833.92436.78140.28942.79648.268
23.09.260 10.19611.68913.09135.17238.07641.63844.18149.728
24.09.886 10.85612.40113.84836.41539.36442.98045.55951.179
25.010.520 11.52413.12014.61137.65240.64644.31446.92852.620
26.011.160 12.19813.84415.37938.88541.92345.64248.29054.052
27.011.808 12.87914.57316.15140.11343.19546.96349.64555.476
28.012.461 13.56515.30816.92841.33744.46148.27850.99356.892
29.013.121 14.25616.04717.70842.55745.72249.58852.33658.301
30.013.787 14.95316.79118.49343.77346.97950.89253.67259.703
40.020.707 22.16424.43326.50955.75859.34263.69166.76673.402
50.027.991 29.70732.35734.76467.50571.42076.15479.49086.661
60.035.534 37.48540.48243.18879.08283.29888.37991.95299.607
70.043.275 45.44248.75851.73990.53195.023100.425104.215112.317
80.051.172 53.54057.15360.391101.879106.629112.329116.321124.839
90.059.196 61.75465.64769.126113.145118.136124.116128.299137.208
100.067.328 70.06574.22277.929124.342129.561135.807140.169149.449
33

34. PHYSICAL AND ASTRONOMICAL CONSTANTS
c Speed of light in vacuo 2 :998‚10
8
m s
€1
e Elementary charge 1 :602‚10
€19
C
mn Neutron rest mass 1 :675‚10
€27
kg
mp Proton rest mass 1 :673‚10
€27
kg
me Electron rest mass 9 :110‚10
€31
kg
h Planck’s constant 6 :626‚10
€34
J s
ˉh Dirac’s constant (=h=2™) 1 :055‚10
€34
J s
k Boltzmann’s constant 1 :381‚10
€23
J K
€1
G Gravitational constant 6 :673‚10
€11
N m
2
kg
€2
› Stefan-Boltzmann constant 5 :670‚10
€8
J m
€2
K
€4
s
€1
c1 First Radiation Constant (= 2™hc
2
) 3:742‚10
€16
J m
2
s
€1
c2 Second Radiation Constant (=hc=k) 1:439‚10
€2
m K
"o Permittivity of free space 8 :854‚10
€12
C
2
N
€1
m
€2
–o Permeability of free scpae 4 ™‚10
€7
H m
€1
NA Avogadro constant 6.022 ‚10
23
mol
€1
R Gas constant 8.314 J K
€1
mol
€1
a0 Bohr radius 5.292 ‚10
€11
m
–B Bohr magneton 9.274 ‚10
€24
J T
€1
‹ Fine structure constant (= 1=137:0) 7.297‚10
€3
MŒ Solar Mass 1.989 ‚10
30
kg
RŒ Solar radius 6.96 ‚10
8
m
LŒ Solar luminosity 3.827 ‚10
26
J s
€1
Mˆ Earth Mass 5.976 ‚10
24
kg
Rˆ Mean earth radius 6.371 ‚10
6
m
1 light year 9.461 ‚10
15
m
1 AU Astronomical Unit 1.496 ‚10
11
m
1 pc Parsec 3.086 ‚10
16
m
1 year 3.156 ‚10
7
s
ENERGY CONVERSION : 1 joule (J) = 6.2415‚10
18
electronvolts (eV)
34