This document contains mathematical formula tables covering a wide range of topics including:
- Greek alphabet
- Indices and logarithms
- Trigonometric, complex number, and hyperbolic identities
- Power series expansions
- Derivatives of common functions
- Integrals of common functions
- Laplace transforms
- And more advanced topics such as vector calculus, mechanics, and statistical distributions.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
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http://www.solohermelin.com.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations visit my website at
http://www.solohermelin.com.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Contemporary communication systems 1st edition mesiya solutions manualto2001
Contemporary Communication Systems 1st Edition Mesiya Solutions Manual
Download:https://goo.gl/DmVRQ4
contemporary communication systems mesiya pdf download
contemporary communication systems mesiya download
contemporary communication systems pdf
contemporary communication systems mesiya solutions
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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 + 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
tan(A ± B) =
sin A sin 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
sin A + sin B = 2 sin A+B cos A−B
2
2
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)
a sin x + b cos x = R sin(x + φ), where R =
If t = tan 1 x then sin x =
2
1
cos x = 2 (eix + e−ix ) ;
eix = cos x + i sin x ;
2t
,
1+t2
cos x =
sin x =
1
2i
√
a2 + b2 and cos φ = a/R, sin φ = b/R.
1−t2
.
1+t2
(eix − e−ix )
e−ix = cos x − i sin x
6. 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 =
√ i(θ+2kπ)/n
n
re
,
k = 0, ±1, ±2, ...
HYPERBOLIC IDENTITIES
cosh x = (ex + e−x ) /2
tanh x = sinh x/ cosh x
sechx = 1/ cosh x
sinh x = (ex − e−x ) /2
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
7. SERIES
Powers of Natural Numbers
n
1
k = n(n + 1) ;
2
k=1
n
n
1
1
k = n(n + 1)(2n + 1);
k 3 = n2 (n + 1)2
6
4
k=1
k=1
2
n−1
Sn =
Arithmetic
(a + kd) =
k=0
n
{2a + (n − 1)d}
2
Geometric (convergent for −1 < r < 1)
n−1
ark =
Sn =
k=0
a(1 − rn )
a
, S∞ =
1−r
1−r
Binomial (convergent for |x| < 1)
(1 + x)n = 1 + nx +
where
n!
n!
x2 + ... +
xr + ...
(n − 2)!2!
(n − r)!r!
n(n − 1)(n − 2)...(n − r + 1)
n!
=
(n − r)!r!
r!
Maclaurin series
xk (k)
x2
f (x) = f (0) + xf (0) + f (0) + ... + f (0) + Rk+1
2!
k!
where Rk+1 =
xk+1 (k+1)
f
(θx), 0 < θ < 1
(k + 1)!
Taylor series
f (a + h) = f (a) + hf (a) +
where Rk+1 =
h2
hk
f (a) + ... + f (k) (a) + Rk+1
2!
k!
hk+1 (k+1)
(a + θh) , 0 < θ < 1.
f
(k + 1)!
OR
f (x) = f (x0 ) + (x − x0 )f (x0 ) +
where Rk+1 =
(x − x0 )2
(x − x0 )k (k)
f (x0 ) + ... +
f (x0 ) + Rk+1
2!
k!
(x − x0 )k+1 (k+1)
(x0 + (x − x0 )θ), 0 < θ < 1
f
(k + 1)!
9. 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
1
x2 − 1
x2
1
−1
10. Product Rule
d
dv
du
(u(x)v(x)) = u(x) + v(x)
dx
dx
dx
Quotient Rule
d
dx
u(x)
v(x)
=
dv
v(x) du − u(x) dx
dx
[v(x)]2
Chain Rule
d
(f (g(x))) = f (g(x)) × g (x)
dx
Leibnitz’s theorem
n(n − 1) (n−2) (2)
n!
dn
.g +...+
f (n−r) .g (r) +...+f.g (n)
(f.g) = f (n) .g+nf (n−1) .g (1) +
f
n
dx
2!
(n − r)!r!
11. INTEGRALS
function
dg(x)
f (x)
dx
xn (n = −1)
f (x)g(x) −
e
ex
sin x
− cos x
1
x
x
cos x
tan x
cosec x
sec x
cot x
1
2 + x2
a
integral
xn+1
n+1
df (x)
g(x)dx
dx
n|x|
Note:- n|x| + K = n|x/x0 |
sin x
n| sec x|
− n| cosec x + cot x|
n tan x
2
or
n| sec x + tan x| = n tan
π
4
+
x
2
n| sin x|
x
1
tan−1
a
a
a2
1
− x2
1 a+x
n
2a a − x
or
x
1
tanh−1
a
a
x2
1
− a2
1 x−a
n
2a x + a
or
−
x
a
(|x| < a)
1
x
coth−1
a
a
1
√
2 − x2
a
sin−1
1
√
2 + x2
a
sinh−1
x
a
or
n x+
1
√
x 2 − a2
sinh x
cosh−1
x
a
or
n|x +
cosh x
cosh x
(|x| > a)
sinh x
tanh x
cosech x
(a > |x|)
√
x 2 + a2
√
x 2 − a2 |
n cosh x
− n |cosechx + cothx|
sech x
2 tan−1 ex
coth x
n| sinh x|
or
n tanh x
2
(|x| > a)
12. Double integral
f (x, y)dxdy =
g(r, s)Jdrds
where
J=
∂(x, y)
=
∂(r, s)
∂x
∂r
∂y
∂r
∂x
∂s
∂y
∂s
13. LAPLACE TRANSFORMS
˜
f (s) =
function
1
tn
eat
sin ωt
cos ωt
sinh ωt
cosh ωt
t sin ωt
∞ −st
f (t)dt
0 e
transform
1
s
n!
n+1
s
1
s−a
ω
s2 + ω 2
s
2 + ω2
s
ω
2 − ω2
s
s
s2 − ω 2
(s2
2ωs
+ 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
14. ˜
Let f (s) = L {f (t)} then
˜
= f (s − a),
L eat f (t)
L {tf (t)} = −
f (t)
t
L
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
dt
d2 y
L
dt2
L
L
t
τ =0
y(τ )dτ
= s˜ − y0 ,
y
= s2 y − sy0 − y0 ,
˜
1
y
˜
s
=
where y0 and y0 are the values of y and dy/dt respectively at t = 0.
Time delay
Let
then
0
g(t) = Ha (t)f (t − a) =
t<a
f (t − a) t > a
˜
L {g(t)} = e−as f (s).
Scale change
L {f (kt)} =
1˜ s
f
.
k
k
Periodic functions
Let f (t) be of period T then
L {f (t)} =
1
1 − e−sT
T
t=0
e−st f (t)dt.
15. Convolution
Let
then
f (t) ∗ g(t) =
t
x=0
f (x)g(t − x)dx =
t
x=0
f (t − x)g(x)dx
˜ g
L {f (t) ∗ g(t)} = f (s)˜(s).
RLC circuit
For a simple RLC circuit with initial charge q0 and initial current i0 ,
1
1
˜
E = r + Ls +
i − Li0 +
q0 .
Cs
Cs
Limiting values
initial value theorem
lim f (t) = s→∞ sf (s),
lim ˜
t→0+
final value theorem
lim f (t) =
t→∞
∞
0
f (t)dt =
provided these limits exist.
˜
lim sf (s),
s→0+
˜
lim f (s)
s→0+
16. Z TRANSFORMS
˜
Z {f (t)} = f (z) =
∞
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
˜
Z {f (t + nT )} = z n f (z) − n−1 z n−k f (kT ) (n > 0)
k=0
Initial value theorem
˜
f (0) = limz→∞ f (z)
17. Final value theorem
˜
f (∞) = lim (z − 1)f (z)
provided f (∞) exists.
z→1
Inverse Formula
f (kT ) =
1
2π
π
−π
˜
eikθ f (eiθ )dθ
FOURIER SERIES AND TRANSFORMS
Fourier series
∞
1
{an cos nωt + bn sin nωt}
f (t) = a0 +
2
n=1
where
2
T
2
=
T
an =
bn
t0 +T
t0
t0 +T
t0
f (t) cos nωt dt
f (t) sin nωt dt
(period T = 2π/ω)
18. Half range Fourier series
sine series
an = 0, bn =
4
T
cosine series
bn = 0, an =
4
T
T /2
0
f (t) sin nωt dt
T /2
0
f (t) cos nωt dt
Finite Fourier transforms
sine transform
˜
fs (n) =
f (t) =
4
T
∞
T /2
0
f (t) sin nωt dt
˜
fs (n) sin nωt
n=1
cosine transform
4 T /2
f (t) cos nωt dt
T 0
∞
1˜
˜
fc (0) +
fc (n) cos nωt
f (t) =
2
n=1
˜
fc (n) =
Fourier integral
1
1
lim f (t) + lim f (t) =
t 0
2 t 0
2π
∞
−∞
eiωt
∞
−∞
f (u)e−iωu du dω
Fourier integral transform
1
˜
f (ω) = F {f (t)} = √
2π
∞
−∞
1
˜
f (t) = F −1 f (ω) = √
2π
e−iωu f (u) du
∞
−∞
˜
eiωt f (ω) dω
19. NUMERICAL FORMULAE
Iteration
Newton Raphson method for refining an approximate root x0 of f (x) = 0
xn+1 = xn −
Particular case to find
√
f (xn )
f (xn )
N use xn+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 , δfn = fn+ 1 − fn− 1
2
2
1
fn = fn − fn−1 , µfn =
f 1 + fn− 1
2
2 n+ 2
Gregory Newton Formula
fp = f0 + p∆f0 +
p!
p(p − 1) 2
∆ f0 + ... +
∆r f0
2!
(p − r)!r!
where p =
x − x0
h
Lagrange’s Formula for n points
n
y=
yi i (x)
i=1
where
i (x)
=
Πn
j=1,j=i (x − xj )
n
Πj=1,j=i (xi − xj )
20. Numerical differentiation
Derivatives at a tabular point
1
1
hf0 = µδf0 − µδ 3 f0 + µδ 5 f0 − ...
6
30
1 4
1 6
h2 f0 = δ 2 f0 − δ f0 + δ f0 − ...
12
90
1 2
1 3
1
1
hf0 = ∆f0 − ∆ f0 + ∆ f0 − ∆4 f0 + ∆5 f0 − ...
2
3
4
5
11 4
5 5
h2 f0 = ∆2 f0 − ∆3 f0 + ∆ f0 − ∆ f0 + ...
12
6
Numerical Integration
x0 +h
T rapeziumRule
x0
fi = f (x0 + ih), E = −
where
h
(f0 + f1 ) + E
2
f (x)dx
h3
f (a), x0 < a < x0 + h
12
Composite Trapezium Rule
x0 +nh
x0
f (x)dx
h2
h
h4
{f0 + 2f1 + 2f2 + ...2fn−1 + fn } − (fn − f0 ) +
(f − f0 )...
2
12
720 n
where f0 = f (x0 ), fn = f (x0 + nh), etc
x0 +2h
Simpson sRule
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)
x0 +nh
x0
f (x)dx
where
h
(f0 + 4f1 + 2f2 + 4f3 + 2f4 + ... + 2fn−2 + 4fn−1 + fn ) + E
3
E=−
nh5 (4)
f (a).
180
x0 < a < x0 + nh
21. Gauss order 1. (Midpoint)
1
−1
f (x)dx = 2f (0) + E
2
E = f (a).
3
where
−1<a<1
Gauss order 2.
1
f (x)dx = f − √ + f
−1
3
1
where
E=
1
√ +E
3
1 v
f (a).
135
−1<a<1
Differential Equations
To solve y = 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 )
h
hK1
K2 = f xn + , yn +
2
2
hK2
h
K3 = f xn + , yn +
2
2
K4 = f (xn + h, yn + hK3 )
22. Chebyshev Polynomials
Tn (x) = cos n(cos−1 x)
To (x) = 1
Un−1 (x) =
T1 (x) = x
Tn (x)
sin [n(cos−1 x)]
√
=
n
1 − x2
Tm (Tn (x)) = Tmn (x).
Tn+1 (x) = 2xTn (x) − Tn−1 (x)
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
1
f (x) = a0 T0 (x) + a1 T1 (x)...aj Tj (x) + ...
2
2 π
aj =
f (cos θ) cos jθdθ
π 0
f (x)dx = constant +A1 T1 (x) + A2 T2 (x) + ...Aj Tj (x) + ...
where Aj = (aj−1 − aj+1 )/2j
j≥1
n≥2
j≥0
23. VECTOR FORMULAE
Scalar product a.b = ab cos θ = a1 b1 + a2 b2 + a3 b3
i
j
k
n
Vector product a × b = ab sin θˆ = a1 a2 a3
b1 b2 b3
= (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 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
≡
grad φ ≡
φ, div A ≡
div grad φ ≡
.(
2
.A, curl A ≡
φ) ≡
div curl A = 0
∂ ∂ ∂
, ,
∂x ∂y ∂z
2
×A
φ (for scalars only)
curl grad φ ≡ 0
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
24. grad (A.B) = A × curl B + B × curl A + (A.
)B + (B.
)A
Integral Theorems
Divergence theorem
surface
A.dS =
volume
div A dV
Stokes’ theorem
surface
( curl A).dS =
contour
A.dr
Green’s theorems
volume
volume
ψ
2
2
φ−φ
2
ψ)dV
=
φ + ( φ)( ψ) dV
=
(ψ
ψ
surface
surface
ψ
∂φ
∂ψ
|dS|
−φ
∂n
∂n
∂φ
|dS|
∂n
where
ˆ
dS = n|dS|
Green’s theorem in the plane
(P dx + Qdy) =
∂Q ∂P
−
∂x
∂y
dxdy
25. 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.
˙
In polar coordinates the velocity is (r, rθ) = rer + rθeθ and the acceleration is
˙ ˙
˙
˙
˙ ¨
¨
¨
r
r − rθ2 , rθ + 2rθ = (¨ − rθ2 )er + (rθ + 2rθ)eθ .
˙˙
˙˙
Centres of mass
The following results are for uniform bodies:
hemispherical shell, radius r
hemisphere, radius r
right circular cone, height h
arc, radius r and angle 2θ
sector, radius r and angle 2θ
1
r
2
3
r
8
3
h
4
from vertex
(r sin θ)/θ
from centre
2
( 3 r sin θ)/θ
from centre
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 .
26. 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 =
tB
tA
F.
dr
dt.
dt
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 λ, µ in F and all a, b, in V ,
ii. λa is defined and is in V
iii. λ(a + b) = λa + λb
28. 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).
29. 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
P (X = r) =
n!
pr (1
(n−r)!r!
r = 0, 1, 2, ..., n
Poisson
P (X = n) =
λ
− p)n−r ,
e−λ λn
,
n!
n = 0, 1, 2, ......
Normal
µ, σ
1
√
σ 2π
f (x) =
exp{− 1
2
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
1
ν
2 1
1
ν
2 1
+ 1 ν2
2
Γ
1
ν
2 2
1
ν1
1
ν12 ν22
x
ν2
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.
34. 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
Permittivity of free space
µo
Permeability of free scpae
NA
Avogadro constant
R
Gas constant
a0
Bohr radius
µB
Bohr magneton
α
Fine structure constant (= 1/137.0)
M
Solar Mass
R
Solar radius
L
Solar luminosity
M⊕
Earth Mass
R⊕
Mean earth radius
1 light year
1 AU
Astronomical Unit
1 pc
Parsec
1 year
1.602 × 10−19 C
1.675 × 10−27 kg
1.673 × 10−27 kg
9.110 × 10−31 kg
6.626 × 10−34 J s
1.055 × 10−34 J s
1.381 × 10−23 J K−1
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
1.496 ×1011 m
3.086 ×1016 m
3.156 ×107 s
ENERGY CONVERSION : 1 joule (J) = 6.2415 × 1018 electronvolts (eV)