This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
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3. GREEK ALPHABET
A a alpha N nu
B ß beta xi
G gamma O o omicron
d delta p pi
E , epsilon P rho
e
Z zeta S s sigma
H eta T t tau
T , theta upsilon
I iota F f phi
,
K kappa X chi
lambda psi
M µ mu omega
INDICES AND LOGARITHMS
am × a n = am + n
(am ) n = a mn
log( AB ) = log A + log B
log( A/B ) = log A - log B
log( A n ) = n log A
a
log b a = log c
logc b
3
4. 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 + cos 2 A = 1
sec2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 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 = cos 2 A - sin 2 A
= 2 cos 2 A - 1
= 1 - 2 sin2 A
tan 2 A = 2tan A
1- tan 2 A
sin 3A = 3 sin A - 4 sin3 A
cos 3A = 4 cos 3 A - 3 cos A
tan 3 A = 3 tan A- tan 3 A
1 - 3tan 2 A
sin A + sin B = 2 sin A +B cos A-B
2 2
4
5. sin A - sin B = 2 cos A+ B sin A-B
2 2
cos A + cos B = 2 cos A+ B 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 )
v
a sin x + b cos x = R sin( x + f ), where R = a 2 + b2 and cos f = a/R , sin f = b/R .
If t = tan 1 x then sin x = 2t , cos x = 1 -t 2 .
2 1+ t2 1+ t2
cos x = 1 ( eix + e-ix ) ; sin x = 1 (eix - -ix )
2 2i
e
eix = cos x + i sin x ; e-ix = cos x - sin x
i
5
6. COMPLEX NUMBERS
v
i= - 1 Note:- ‘ j ’ often used rather than ‘ i’.
Exponential Notation
ei = cos + i sin
De Moivre’s theorem
[r (cos + i sin )]n = rn (cos n + i sin n )
nth roots of complex numbers
If z = r i = r (cos + i sin ) then
e
vr
z 1/n = ne
i ( +2 kp ) /n , = 0 , ± 1, ± 2, ...
k
HYPERBOLIC IDENTITIES
cosh x = ( ex + e-x ) / 2 sinh x = ( ex - -x )/2
e
tanh x = sinh x cosh x
/
sech x = 1 / cosh x cosech x = 1 / sinh x
coth x = cosh x sinh x = 1 / tanh x
/
cosh i = cos x sinh i = i sin x
x x
cos i = cosh x sin i = i sinh x
x x
cosh 2 A - sinh 2 A = 1
sec 2 A = 1 - tanh 2 A
h
cosec 2 A = coth 2 A - 1
h
6
7. SERIES
Powers of Natural Numbers
n n n
k = 1 n ( n + 1) ; k = 1 n ( n + 1)(2 n + 1);
2 k3 = 1 n 2 (n + 1) 2
2 6 4
k =1 k =1 k =1
n- 1 n
Arithmetic Sn = (a + k ) = { 2a + ( n - 1)d}
d 2
k =0
Geometric (convergent for - 1 < r < 1)
n- 1 a(1 - n ) a
Sn = a k = =
r 1rr,S
- 8
1-
k =0
r
Binomial (convergent for |x| 1)
<
n! n!
(1 + x )n = 1 + n + x 2 + ... + x r + ...
x ( n - 2)!2! ( n - r )!r!
n! n ( n - 1)( n - 2)... ( n - r + 1)
where
(n - r )!r ! = r!
Maclaurin series
x2 xk
f (x ) = f (0) + x (0) + f (0) + ... + f ( k ) (0) + R k +1
f 2! k!
x k +1
where R k +1 = f ( k +1) ( x ) , 0 < < 1
( k + 1)!
Taylor series
h2 hk
f ( a + h ) = f ( a) + hf ( a) + f ( a) + ... + f ( k ) (a ) + R k +1
2! k!
h k +1
where R k +1 = f ( k +1) (a + h ) , 0 < < 1.
(k + 1)!
OR
x - 0 )2 x - 0 )k ( k )
f (x ) = f ( x0 ) + ( x - 0 ) f (x 0 ) + ( x f (x 0 ) + ... + ( x f (x 0 ) + R k +1
x 2! k!
x - 0 ) k +1 ( k +1)
where R k +1 = ( x f (x 0 + ( x - 0
) ), 0 < < 1
( k + 1)! x
7
8. Special Power Series
x2 x3 xr
e =1+ x +
x ... + ... (all x )
2! + 3! + r! +
3 x5 7 - 1)r x2 r +1
sin x = x - - ... + ( ... (all x )
x 3! + 5! x 7! + (2 r + 1)! +
x4 62 - 1) r x 2r
cos x = 1 - - ... + ( ... (all x )
x 2! + 4! x 6! + (2r )! +
x3 x5 x7
tan x = x + ... ( |x| p )
3 + 2 15 + 17315 + <
2
x3 .3 x5 .3.5 x7
sin - 1 x = x +1
2 3 + 1 2. 4 5 + 1 2.4.6 7 +
.3.5.... (2 n - 1) x 2n +1
... + 1 ... ( |x| 1)
2.4. 6.... (2 n ) 2n + 1 + <
x5 3 7 x2 n +1
tan - 1 x = x - - ... + ( - 1) n ... ( |x| 1)
x 3+ 5 x 7+ 2n + 1 + <
x3 2 4 xn
n (1 + x ) = x - - ... + ( - 1) n +1 + ... ( - 1 < x = 1)
x 2+ 3 x 4+ n
x3 x5 x7 x 2 n +1
sinh x = x + ... + ... (all x )
3! + 5! + 7! + (2n + 1)! +
x2 x4 x6 x 2n
cosh x = 1 + ... + ... (all x )
2! + 4! + 6! + (2n )! +
x5 17x 7
3
tanh x = x - - ... ( |x| p )
x 3 + 2 15 315 + <
2
1 x3 .3 x5 1.3.5 x 7
sinh - 1 x = x -
- 2 3 + 1 2.4 5 2.4.6 7 +
1.3.5... (2n - 1) x2n +1
... + ( - 1) n ... ( |x| 1)
2.4.6... 2n 2n + 1 + <
x3 x5 x7 2 n +1
tanh - 1 x = x+ ... x ... ( |x| 1)
3+ 5+ 7+ 2n + 1 + <
8
9. DERIVATIVES
function derivative
xn n n- 1
xx
ex e
ax ( a > 0) ax na
n 1
x
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
cot x - cosec 2 x
1
sin- 1 x v
1- 2
x
1
cos- 1 x - v
1- 2
x
1
tan - 1 x
1 + x2
sinh x cosh x
cosh x sinh x
tanh x sech 2 x
cosech x - cosech x coth x
sech x - sech x tanh x
coth x - cosech 2 x
1
sinh - 1
x v
1 + x2
cosh - 1
x (x > 1) 1 v
x 2 - 1
tanh - 1 x ( |x| 1) 1
< 1- 2
x
1
coth - 1 x (|x| 1) -
> x2 - 1
9
10. Product Rule
d d du
( u( x )v ( x )) = u ( x) v + v( x )
d d d
x x x
Quotient Rule
d u (x ) v ( x ) du - u ( x ) dv
dx dx
d v(x ) = [v ( x)]2
x
Chain Rule
d
(f ( g( x))) = f (g( x )) × g (x )
d
x
Leibnitz’s theorem
dn n (n - 1) n!
( f.g ) = f ( n ) .g + nf ( n- 1) .g(1) + f ( n- 2) .g (2) + ... + f ( n-r ) .g( r ) + ... + f .g ( n )
d n 2! ( n - r )!r!
x
10
11. INTEGRALS
function integral
dg(x ) (x )
f (x ) (x ) g(x ) - df ( x )d
dx f dx g x
xn (n = - 1) xn +1
n +1
1 n| Note:- n| + K = n|x/ 0 |
x
x| x| x
ex ex
sin x cos x
-
cos x sin x
tan x n| sec x
|
cosec x - n| cosec x + cot x or n tan x
2
|
sec x n| sec x + tan x = n tan p + x
4 2
|
cot x n| sin x
1 1 | x
tan - 1
a2 + x 2 a a
1 1 + x x
or 1 tanh - 1 ( |x| < a )
a2 - 2 2a n aa - x a a
x
1 1 1 x
or - coth - 1 (|x| > a )
x2 - a 2 2a n xx- + a
a a a
1 x
v sin- 1 ( a > |x| )
a 2 - 2 a
x
1 x v
v sinh - 1
or nx + x 2 + a2
a 2 + x2 a
1 x v
v cosh - 1 or n| + x 2 -a 2 | ( |x| > a )
x 2 -a 2 a x
sinh x cosh x
cosh x sinh x
tanh x n cosh x
cosech x - n |cosechx + cothx| or n tanh x
2
sech x 2 tan - 1 ex
coth x n| sinh x
|
11
12. Double integral
f ( x, ) dxdy = g( r, ) J drds
y s
where
(x, ) x x
J = r s
(y ) =
r, y y
s r s
12
13. LAPLACE TRANSFORMS
f˜ ( s) = 0
8 e- f (t) dt
st
function transform
11
s
n!
tn
s n +1
1
eat
s-a
sin t
s2 + 2
cos t
s s2 + 2
sinh t
s2 - 2
cosh t
s s2 - 2
2s
t sin t
( s2 + )
2 2
-
2 2
t cos t
s ( s2 + )
2 2
e-as
H a ( t) = H ( t - a )
s
d( t) 1
n!
eat tn
( s - a ) n +1
eat sin t
( s - a )2 + 2
eat cos t s - a
( s - a )2 + 2
eat sinh t
( s - a )2 - 2
eat cosh t s - a
( s - a )2 - 2
13
14. Let ˜f (s ) = L {f ( t) } then
Le at f (t) = ˜ f (s - a ) ,
L {tf ( t) } = - d ( f˜ ( s)) ,
d
( t) 8s
Lf = f˜ (x ) d if this exists .
t x=s x
Derivatives and integrals
Let y = y (t ) and let ˜ y = L {y (t )} then
˜ L dy = s 0 ,
y-y dt
2 y
Ld = s2 y -
˜ 0
- 0
,
dt 2
sy y
t
L y ( t ) dt =1 y
˜
t =0 s
where y0 and y0 are the values of y and dy/dt respectively at t = 0.
Time delay
0 t<a
Let g( t) = H a ( t)f (t - a ) =
f (t - a ) t > a
then L {g ( t) } = e-as f˜ ( s ).
Scale change
L {f ( k ) } = 1 f˜
t ks k
.
Periodic functions
Let f (t) be of period T then
T
L {f ( t) } = 1 e- f ( t) dt.
1- - t =0
st
sT
e
14
15. Convolution
Let f ( t) * g ( t) = t
x =0
f ( x ) g(t - )d = t
x =0
f (t - )g( x )d
x x x x
then L {f (t ) * g( t) } = ˜f (s )˜ ( s).
g
RLC circuit
For a simple RLC circuit with initial charge q0 and initial current i0 ,
˜
E = r + L +1 +1 .
0
s Cs i - Li Cs q0
Limiting values
initial value theorem
lim f ( t) = lim s f˜ (s ),
t 0+ s8
final value theorem
lim f (t ) = lim s f˜ ( s ) ,
t8 s 0+
8
f ( t) dt = lim f˜ ( s)
0 s 0+
provided these limits exist.
15
16. Z TRANSFORMS
8
Z {f ( t)} = ˜f ( z ) = f ( k ) z-k
k =0 T
function transform
dt,nT z -n ( n = 0)
z
e-at
z - -aT
e
T -aT
t -at
e ( z ze- -aT ) 2
e
T 2 z -aT ( z + e-aT )
t2 e-at e (z - -aT ) 3
e
sinh aT
sinh at z
z 2 - 2z cosh aT + 1
( z cosh aT )
cosh at z
z 2 - -2z cosh aT + 1
-aT sin T
e-at sin t
ze z 2 - 2z -aT cos T + e- 2aT
e
(z - -aT cos T )
e-at cos t e
z z 2 - 2z -aT cos T + e- 2aT
e
-aT (z 2 - - 2 aT ) sin T
t -at sin t T ze
e ( z 2 - 2z -aT e T + e- 2 aT )2
cos
e
-aT (z 2 cos T - 2z -aT + e- 2 aT cos T )
t -at cos t T ze e
e ( z 2 - 2z -aT cos T + e- 2 aT )2
e
Shift Theorem
Z {f (t + nT ) } = z n f˜ ( z ) - n- 1
k =0
z n-k f (k ) ( n > 0)
T
Initial value theorem
f (0) = lim z8 f˜ ( z )
16
17. Final value theorem
f ( 8 ) = lim (z 1) f˜ ( z ) provided f (8 ) exists
z 1
- .
Inverse Formula
p
f (k ) = 1 eik f˜ ( ei )d
T 2p -p
FOURIER SERIES AND TRANSFORMS
Fourier series
8
f ( t) = 1 a0 + {a n cos n t + bn sin n t} (period T = 2 p/ )
2
n =1
where
t0 + T
an =2 f ( t) cos n t dt
T t0
t0 + T
bn =2 f ( t) sin n t dt
T t0
17
18. Half range Fourier series
T 2
sine series n = 0 ,bn = 4 / f ( t) sin n t dt
a T 0
T 2
cosine series n
=0 ,an =4 / f ( t) cos n t dt
b T 0
Finite Fourier transforms
sine transform
T 2
f˜s ( n ) = 4 / f ( t) sin n t dt
T 0
8
f ( t) = f˜s (n ) sin n t
n =1
cosine transform
T 2
f˜c ( n ) = 4 / f (t ) cos n t dt
T 0
8
f (t ) = 1 f˜c (0) + f˜c (n ) cos n t
2 n =1
Fourier integral
1 8
f (t) + lim f ( t ) = 1 ei t 8 f ( u) e-i u du d
2 lim
t 0 t 0 2p -8 -8
Fourier integral transform
8
f˜ ( ) = F {f (t )} = 1v e-i u f ( u) du
2p -8
8
f ( t) = F- 1 f˜ ( ) = 1 v ei t f˜ ( ) d
2p -8
18
19. NUMERICAL FORMULAE
Iteration
Newton Raphson method for refining an approximate root x 0 of f (x ) = 0
(xn )
x n +1 = xn -
f f (x n )
v
Particular case to find N use x n +1 = 1 xn + N .
2 xn
Secant Method
(x n ) - (x n- 1 )
x n +1 = x n - ( xn ) /
f f xn f - n- 1
x
Interpolation
f n = f n +1 - n
, df n
= fn+ 1
- n- 1
f 2
f 2
f n = f n - n- 1
, µf n
= 1 fn+ 1
+ f n- 1
f 2 2 2
Gregory Newton Formula
p(p - 1) p!
fp = f0 + p f0 + 2 f 0 + ... + r f0
2! ( p - r )!r!
x - 0
where p = x
h
Lagrange’s Formula for n points
n
y= yi i (x )
i =1
where
n
j =1 ,j = i
(x - j )
i (x ) = x
n
j =1 ,j = i
( xi - j
)
x
19
20. Numerical di erentiation
Derivatives at a tabular point
1 3
hf 0 = µdf 0 - µd f 0 + 1 µd5 f 0 - ...
6 30
1 4
h2 f 0 = d2 f 0 - d f 0 + 1 d6 f 0 - ...
12 90
1 2 1 4
hf 0 = f0 - f0 + 1 3f0 - f0 + 1 5 f 0 - ...
2 3 4 5
5 5
h2 f 0 = 2 f -
0
3 f + 11
0
4 f
0
- f 0 + ...
12 6
Numerical Integration
x0 + h
T rapeziumRule f ( x ) dx h f0 + f1 ) + E
x0 2(
3
where f i
= f (x 0 + ih ), E = -h f ( a), 0
<a<x 0
+ h
12 x
Composite Trapezium Rule
x0 + nh 2 h4
f (x )dx h { 0
+ 2 f 1 + 2 f 2 + ...2f n- 1
+ fn } - h f - )+ f - ) ...
x0 2f 12 ( n f 0 720 ( n f 0
where f 0 = f (x 0 ), f n
= f ( x0 + nh ), etc
x 0 +2 h
Simpson sRule f ( x) dx h f 0 +4 f 1 + f 2 )+ E
x0 3(
5
wher E = -h f (4) ( a) x0 < a < x 0
+ 2 h.
e 90
Composite Simpson’s Rule ( n even)
x0 + nh
f (x )dx h f 0 + 4 f 1 + 2 f 2 + 4 f 3 + 2 f 4 + ... + 2 f n- 2 + 4 f n- 1 + fn ) + E
x0 3(
5
where E = -nh f (4) ( a) . 0
<a<x 0
+ nh
180 x
20
21. Gauss order 1. (Midpoint)
1
f (x )d = 2 f (0) + E
- 1 x
where E = 2 f (a) . - 1 <a< 1
3
Gauss order 2.
1 1 1
f (x ) d = f v f v E
- 1 x - 3+ 3+
where E =1 f v ( a) . - 1<a< 1
135
Di erential Equations
To solve y = f ( x, ) given initial condition y0 at x0 , n
= x 0 + nh .
y x
Euler’s forward method
yn +1 = yn + hf (x n , n ) n = 0 , 1, 2, ...
y
Euler’s backward method
yn +1 = yn + hf ( xn +1 , n +1 ) n = 0 , 1, 2, ...
y
Heun’s metho d (Runge Kutta order 2)
h
yn +1 = yn + f (x n , n
) + f ( xn + h, y n + hf (x n , n
))) .
2( y y
Runge Kutta order 4.
h
yn +1 = yn + K + 2 K2 + 2 K3 + K4 )
6( 1
where
K1 = f (x n , n )
y h hK 1
K2 = f n
+ , n
+
x 2y 2
h hK 2
K3 = f n
+ , n
+
x 2y 2
K4 = f (x n + h, y n + hK 3 )
21
22. Chebyshev Polynomials
Tn (x ) = cos n (cos - 1 x)
To (x ) = 1 T1 ( x ) = x
Tn (x ) n (cos - 1 x)]
Un- 1
(x ) = = sin [ v
n 1- 2
x
Tm (Tn (x )) = Tmn ( x ) .
Tn +1 ( x ) = 2 x n ( x ) - T n- 1 (x )
T
Un +1 ( x ) = 2 x n ( x) - U n- 1 (x )
U T (x ) (x )
n +1
Tn ( x )d =1 - T n- 1 constant, n = 2
x 2 n +1 n- 1+
f ( x ) = 1 a0 T0 ( x ) + a1 T1 ( x ) ...a j Tj ( x) + ...
2
p
where a j
=2 f (cos ) cos j d j = 0
p 0
and f ( x ) d = constant + A 1 T1 ( x ) + A2 T2 (x ) + ...A j Tj ( x ) + ...
x
where A j = ( aj- 1
-a j +1
) / 2j j = 1
22
23. VECTOR FORMULAE
Scalar product a .b = ab cos = a1 b1 + a2 b2 + a3 b3
ijk
Vector product a × b = ab sin n =
ˆ a1 a2 a3
b1 b2 b3
= ( a2 b3 - a 3 2
b )i + ( a3 b1 - a 1 b3 )j + ( a1 b2 - a 2
b1 )k
Triple products
a1 a2 a3
[a , b , c] = (a × b) .c = a .(b × c) = b1 b2 b3
c1 c2 c3
a × (b × c) = (a .c)b - (a .b)c
Vector Calculus
=
x y z
, ,
grad f = f, div A = . A , curl A = × A
div grad f=. ( f)= 2 f (for scalars only)
div curl A = 0 curl grad f= 0
2 A = grad div A - curl curl A
(aß ) = a ß + ßa
div ( a A) = a div A + A . ( a )
curl ( a A) = a curl A - A × ( a )
div (A × B) = B . curl A - A . curl B
curl (A × B) = A div B - B div A + (B . )A - (A . )B
23
24. grad (A .B) = A × curl B + B × curl A + (A . )B + (B . )A
Integral Theorems
Divergence theorem
A .dS = div A dV
surface volume
Stokes’ theorem
( curl A) .dS = A .dr
surface contour
Green’s theorems
( 2 f - 2 )dV = f S|
volume f surface n-f n |d
2 f +( f )( ) dV = f S|
volume surface n |d
where
dS = ˆ n |dS |
Green’s theorem in the plane
Q
( P dx + Qdy ) =
x - P y dxdy
24
25. MECHANICS
Kinematics
Motion constant acceleration
v=u+f t s = u t + 1 t2 = 1 t
, 2f 2 (u + v)
v2 = u 2 + 2f .s
General solution of d2 x = - 2 x is
dt 2
x = a cos t + b sin t = R sin( t + f)
v
where R = a 2 + b and cos f = a/R , sin f = b/R .
2
In polar coordinates the velocity is ( r, )= rer + r e and the acceleration is
r
¨
r -
¨ 2 , +2 r = (¨r - 2 )e r + ( r ¨ + 2 r )e .
r r r
Centres of mass
The following results are for uniform bodies:
hemispherical shell, radius r 1 r from centre
2
hemisphere, radius r 3 r from centre
8
right circular cone, height h 3 h from vertex
4
arc, radius r and angle 2 ( r sin )/ from centre
sector, radius r and angle 2 ( 2 r sin )/ from centre
3
Moments of inertia
i. The moment of inertia of a body of mass m about an axis = I + mh 2 , 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 I 1 and I 2 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
26. iii. The following moments of inertia are for uniform bodies about the axes stated:
ro d, length , through mid-point, perpendicular to rod 1 m 2
12
ho op, radius r , through centre, perpendicular to ho op m 2
r
disc, radius r, through centre, perpendicular to disc 1 m 2
2
r
sphere, radius r, diameter 2 m 2
5
r
Work done
tB r
W = F. d
tA dt dt.
26
27. 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 ,µ in F and all a, b, in V ,
ii. a is defined and is in V
iii. (a + b) = a + b
27
28. iv. ( + µ)a = a + µa
v. ( .µ )a = (µa )
vi. if 1 is an element of F such that 1 . = for all in F , then 1 a = 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 re extive)
ii. aRb bRa (R is symmetric)
iii. ( aRb and bRc ) aRc ( R is transitive).
28
29. PROBABILITY DISTRIBUTIONS
Name Parameters Probability distribution / Mean Variance
density function
Binomial n, p P (X = r) = n! pr (1 - p )n-r , np np (1 - p )
( n-r )! r !
r = 0 , 1, 2, ..., n
Poisson P (X = n) = e- n
,
n!
n = 0 , 1, 2, ......
2
Normal µ, s f (x ) = v1 exp { 1 x-
µs
}, µ s 2
s 2p 2
-
-8 < x < 8
Exponential f (x ) = e -x , 1 1
2
x > 0, > 0
THE F -DISTRIBUTION
The function tabulated on the next page is the inverse cumulative distribution
function of Fisher’s F -distribution having 1
and 2
degrees of freedom. It is defined
by
1
1
+ 1
2 1 1 x
2 2
P =G 2
1
1 2
2
2
u1
2 1 - 1 ( 2
+ 1
u) - 1
2
( 1 + 2 ) du.
G 1
1 G 1
2 0
2 2
If X has an F -distribution with 1
and 2
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
34. PHYSICAL AND ASTRONOMICAL CONSTANTS
c Speed of light in vacuo 2 .998 × 108 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 Js
¯ Dirac’s constant (= h/ 2p ) 1 .055 × 10- 34 Js
h
k Boltzmann’s constant 1 .381 × 10- 23 JK- 1
G Gravitational constant 6 .673 × 10- 11 N m 2 kg- 2
s Stefan-Boltzmann constant 5 .670 × 10- 8 Jm - 2 K- 4 s- 1
c1 First Radiation Constant (= 2 phc 2 ) 3 .742 × 10- 16 J m 2 s- 1
c2 Second Radiation Constant (= hc/ ) 1 .439 × 10- 2 mK
k
eo Permittivity of free space 8 .854 × 10- 12 C2 N- 1 m- 2
µo Permeability of free scpae 4 p × 10- 7 H m- 1
NA Avogadro constant 6.022 × 1023 mol- 1
R Gas constant 8.314 J K - 1 mol - 1
a0 Bohr radius 5.292 × 10- 11 m
µB Bohr magneton 9.274 × 10- 24 JT - 1
a Fine structure constant (= 1 / 137. 0) 7.297 × 10- 3
M Solar Mass 1.989 × 1030 kg
R Solar radius 6.96 × 108 m
L Solar luminosity 3.827 × 1026 J s - 1
M Earth Mass 5.976 × 1024 kg
R Mean earth radius 6.371 × 106 m
1 light year 9.461 × 1015 m
1 AU Astronomical Unit 1.496 × 1011 m
1 pc Parsec 3.086 × 1016 m
1 year 3.156 × 107 s
ENERGY CONVERSION : 1 joule (J) = 6.2415 × 1018 electronvolts (eV)
34