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Department of Mathematics, UMIST
MATHEMATICAL FORMULA TABLES
Version 2
...


1−t2

...


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,
...
+
xr +
...
(n − r + 1)
n!
=
(n − r)!r!
r!

Maclaurin series
xk (k)
x2
f (x) = f (0) + xf (0) + f (0) +
...
+ f (k) (a) + Rk+1
2!
k!

hk+1 (k+1)
(a + θh) , 0 < θ < 1
...
+
f (x0 ) + Rk+1
2!
k!

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

Special Power Series

x2 x3
xr
+
...
+
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 −

(−1)r x2r
x2 x4 x6
+
−
+
...

2!
4!
6!
(2r)!

(all x)

tan x = x +

x3 2x5 17x7
+
+
+
...
3 x5 1
...
5 x7
+
+
+
2 3
2
...
4
...
+

tan−1 x = x −

1
...
5
...

2
...
6
...
+ (−1)n
+
...
+ (−1)n+1 +
...
+
+
...
+
+
...

3
15
315

(|x| < π )
2

sinh−1 x = x −

1 x3 1
...
3
...
4 5
2
...
6 7


...
3
...
(2n − 1) x2n+1
+
...
4
...
2n 2n + 1

x2n+1
x3 x5 x7
+
+
+
...

3
5
7
2n + 1

(|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

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

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

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

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)

Double integral
f (x, y)dxdy =

g(r, s)Jdrds

where
J=

∂(x, y)
=
∂(r, s)

∂x
∂r
∂y
∂r

∂x
∂s
∂y
∂s

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

˜
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
...

Time delay
Let
then





0

g(t) = Ha (t)f (t − a) = 



t
f (t − a) t > a

˜
L {g(t)} = e−as f (s)
...

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
...


RLC circuit
For a simple RLC circuit with initial charge q0 and initial current i0 ,
1
1
˜
E = r + Ls +
i − Li0 +
q0
...


˜
lim sf (s),

s→0+

˜
lim f (s)

s→0+

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)

Final value theorem
˜
f (∞) = lim (z − 1)f (z)

provided f (∞) exists
...


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 +
...

6
30
1 4
1 6
h2 f0 = δ 2 f0 − δ f0 + δ f0 −
...

2
3
4
5
11 4
5 5
h2 f0 = ∆2 f0 − ∆3 f0 + ∆ f0 − ∆ f0 +
...
2fn−1 + fn } − (fn − f0 ) +
(f − f0 )
...


Composite Simpson’s Rule (n even)
x0 +nh
x0

f (x)dx

where

h
(f0 + 4f1 + 2f2 + 4f3 + 2f4 +
...

180

x0 < a < x0 + nh

Gauss order 1
...

3

where

−1
Gauss order 2
...

135

−1
Differential Equations
To solve y = f (x, y) given initial condition y0 at x0 , xn = x0 + nh
...


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

n = 0, 1, 2,
...

2
Runge Kutta order 4
...

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)
...

2
2 π
aj =
f (cos θ) cos jθdθ
π 0

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


where Aj = (aj−1 − aj+1 )/2j

j≥1

n≥2

j≥0

VECTOR FORMULAE

Scalar product a
...
c = a
...
c)b − (a
...
(

2


...
( α)
curl (αA) = α curl A − A × ( α)
div (A × B) = B
...
curl B
curl (A × B) = A div B − B div A + (B
...


)B

grad (A
...


)B + (B
...
dS =

volume

div A dV

Stokes’ theorem
surface

( curl 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

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

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


˙
In polar coordinates the velocity is (r, rθ) = rer + rθeθ and the acceleration is
˙ ˙
˙
˙
˙ ¨
¨
¨
r
r − rθ2 , rθ + 2rθ = (¨ − rθ2 )er + (rθ + 2rθ)eθ
...
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
...
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
...
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
...

dt

ALGEBRAIC STRUCTURES

A group G is a set of elements {a, b, c,
...
a ∗ b is in G for all a, b in G
ii
...
G contains an element e, called the identity element, such that e ∗ a = a = a ∗ e
for all a in G

iv
...

A commutative (or Abelian) group is one for which a ∗ b = b ∗ a for all a, b, in G
...
} — with two binary operations + and
...
F is a commutative group with respect to + with identity 0
ii
...
with
identity 1
iii
...
(b + c) = a
...
c for all a, b, c, in F
...
} — with a binary

operation + such that

i
...
λa is defined and is in V
iii
...
(λ + µ)a = λa + µa
v
...
if 1 is an element of F such that 1
...


An equivalence relation R between the elements {a, b, c,
...
aRa (R is reflextive)
ii
...
(aRb and bRc) ⇒ aRc (R is transitive)
...
, n
Poisson

P (X = n) =

λ

− p)n−r ,

e−λ λn
,
n!

n = 0, 1, 2,
...
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
...
(X ≤ x) = P
...
95, P = 0
...
99, the upper number
in each set being the value for P = 0
...


ν2 ν1 : 1 ν1 : 2

3

4

5

6

7

8

9

10

12

15

20

25

50 100

161 199 216 225 230 234 237 239 241 242 244 246 248 249 252 253
648 799 864 900 922 937 948 957 963 969 977 985 993 998 1008 1013 1
4052 5000 5403 5625 5764 5859 5928 5981 6022 6056 6106 6157 6209 6240 6303 6334
18
...
00 19
...
25 19
...
33 19
...
37 19
...
40 19
...
43 19
...
46 19
...
49
2 38
...
00 39
...
25 39
...
33 39
...
37 39
...
40 39
...
43 39
...
46 39
...
49 2
98
...
00 99
...
25 99
...
33 99
...
37 99
...
40 99
...
43 99
...
46 99
...
49
10
...
55 9
...
12 9
...
94 8
...
85 8
...
79 8
...
70 8
...
63 8
...
55
3 17
...
04 15
...
10 14
...
73 14
...
54 14
...
42 14
...
25 14
...
12 14
...
96 3
34
...
82 29
...
71 28
...
91 27
...
49 27
...
23 27
...
87 26
...
58 26
...
24
7
...
94 6
...
39 6
...
16 6
...
04 6
...
96 5
...
86 5
...
77 5
...
66
4 12
...
65 9
...
60 9
...
20 9
...
98 8
...
84 8
...
66 8
...
50 8
...
32 4
21
...
00 16
...
98 15
...
21 14
...
80 14
...
55 14
...
20 14
...
91 13
...
58
6
...
79 5
...
19 5
...
95 4
...
82 4
...
74 4
...
62 4
...
52 4
...
41
5 10
...
43 7
...
39 7
...
98 6
...
76 6
...
62 6
...
43 6
...
27 6
...
08
5
16
...
27 12
...
39 10
...
67 10
...
29 10
...
05 9
...
72 9
...
45 9
...
13
5
...
14 4
...
53 4
...
28 4
...
15 4
...
06 4
...
94 3
...
83 3
...
71
6 8
...
26 6
...
23 5
...
82 5
...
60 5
...
46 5
...
27 5
...
11 4
...
92
6
13
...
92 9
...
15 8
...
47 8
...
10 7
...
87 7
...
56 7
...
30 7
...
99
5
...
74 4
...
12 3
...
87 3
...
73 3
...
64 3
...
51 3
...
40 3
...
27
7 8
...
54 5
...
52 5
...
12 4
...
90 4
...
76 4
...
57 4
...
40 4
...
21
7
12
...
55 8
...
85 7
...
19 6
...
84 6
...
62 6
...
31 6
...
06 5
...
75
5
...
46 4
...
84 3
...
58 3
...
44 3
...
35 3
...
22 3
...
11 3
...
97
8 7
...
06 5
...
05 4
...
65 4
...
43 4
...
30 4
...
10 4
...
94 3
...
74
8
11
...
65 7
...
01 6
...
37 6
...
03 5
...
81 5
...
52 5
...
26 5
...
96
5
...
26 3
...
63 3
...
37 3
...
23 3
...
14 3
...
01 2
...
89 2
...
76
9 7
...
71 5
...
72 4
...
32 4
...
10 4
...
96 3
...
77 3
...
60 3
...
40
9
10
...
02 6
...
42 6
...
80 5
...
47 5
...
26 5
...
96 4
...
71 4
...
41
4
...
10 3
...
48 3
...
22 3
...
07 3
...
98 2
...
85 2
...
73 2
...
59
10 6
...
46 4
...
47 4
...
07 3
...
85 3
...
72 3
...
52 3
...
35 3
...
15 10
10
...
56 6
...
99 5
...
39 5
...
06 4
...
85 4
...
56 4
...
31 4
...
01
4
...
89 3
...
26 3
...
00 2
...
85 2
...
75 2
...
62 2
...
50 2
...
35
12 6
...
10 4
...
12 3
...
73 3
...
51 3
...
37 3
...
18 3
...
01 2
...
80 12
9
...
93 5
...
41 5
...
82 4
...
50 4
...
30 4
...
01 3
...
76 3
...
47
4
...
68 3
...
06 2
...
79 2
...
64 2
...
54 2
...
40 2
...
28 2
...
12
15 6
...
77 4
...
80 3
...
41 3
...
20 3
...
06 2
...
86 2
...
69 2
...
47 15
8
...
36 5
...
89 4
...
32 4
...
00 3
...
80 3
...
52 3
...
28 3
...
98
4
...
49 3
...
87 2
...
60 2
...
45 2
...
35 2
...
20 2
...
07 1
...
91
20 5
...
46 3
...
51 3
...
13 3
...
91 2
...
77 2
...
57 2
...
40 2
...
17 20
8
...
85 4
...
43 4
...
87 3
...
56 3
...
37 3
...
09 2
...
84 2
...
54
4
...
39 2
...
76 2
...
49 2
...
34 2
...
24 2
...
09 2
...
96 1
...
78
25 5
...
29 3
...
35 3
...
97 2
...
75 2
...
61 2
...
41 2
...
23 2
...
00 25
7
...
57 4
...
18 3
...
63 3
...
32 3
...
13 2
...
85 2
...
60 2
...
29
4
...
18 2
...
56 2
...
29 2
...
13 2
...
03 1
...
87 1
...
73 1
...
52
50 5
...
97 3
...
05 2
...
67 2
...
46 2
...
32 2
...
11 1
...
92 1
...
66 50
7
...
06 4
...
72 3
...
19 3
...
89 2
...
70 2
...
42 2
...
17 1
...
82
3
...
09 2
...
46 2
...
19 2
...
03 1
...
93 1
...
77 1
...
62 1
...
39
100 5
...
83 3
...
92 2
...
54 2
...
32 2
...
18 2
...
97 1
...
77 1
...
48 100
6
...
82 3
...
51 3
...
99 2
...
69 2
...
50 2
...
22 2
...
97 1
...
60
1

NORMAL DISTRIBUTION
The function tabulated is the cumulative distribution function of a standard N (0, 1)
random variable, namely
x
1 2
1
Φ(x) = √
e− 2 t dt
...
(X ≤ x)
...
00

0
...
02

0
...
04

0
...
06

0
...
08

0
...
0
0
...
2
0
...
4

0
...
5398
0
...
6179
0
...
5040
0
...
5832
0
...
6591

0
...
5478
0
...
6255
0
...
5120
0
...
5910
0
...
6664

0
...
5557
0
...
6331
0
...
5199
0
...
5987
0
...
6736

0
...
5636
0
...
6406
0
...
5279
0
...
6064
0
...
6808

0
...
5714
0
...
6480
0
...
5359
0
...
6141
0
...
6879

0
...
6
0
...
8
0
...
6915
0
...
7580
0
...
8159

0
...
7291
0
...
7910
0
...
6985
0
...
7642
0
...
8212

0
...
7357
0
...
7967
0
...
7054
0
...
7704
0
...
8264

0
...
7422
0
...
8023
0
...
7123
0
...
7764
0
...
8315

0
...
7486
0
...
8078
0
...
7190
0
...
7823
0
...
8365

0
...
7549
0
...
8133
0
...
0
1
...
2
1
...
4

0
...
8643
0
...
9032
0
...
8438
0
...
8869
0
...
9207

0
...
8686
0
...
9066
0
...
8485
0
...
8907
0
...
9236

0
...
8729
0
...
9099
0
...
8531
0
...
8944
0
...
9265

0
...
8770
0
...
9131
0
...
8577
0
...
8980
0
...
9292

0
...
8810
0
...
9162
0
...
8621
0
...
9015
0
...
9319

1
...
6
1
...
8
1
...
9332
0
...
9554
0
...
9713

0
...
9463
0
...
9649
0
...
9357
0
...
9573
0
...
9726

0
...
9484
0
...
9664
0
...
9382
0
...
9591
0
...
9738

0
...
9505
0
...
9678
0
...
9406
0
...
9608
0
...
9750

0
...
9525
0
...
9693
0
...
9429
0
...
9625
0
...
9761

0
...
9545
0
...
9706
0
...
0
2
...
2
2
...
4

0
...
9821
0
...
9893
0
...
9778
0
...
9864
0
...
9920

0
...
9830
0
...
9898
0
...
9788
0
...
9871
0
...
9925

0
...
9838
0
...
9904
0
...
9798
0
...
9878
0
...
9929

0
...
9846
0
...
9909
0
...
9808
0
...
9884
0
...
9932

0
...
9854
0
...
9913
0
...
9817
0
...
9890
0
...
9936

2
...
6
2
...
8
2
...
9938
0
...
9965
0
...
9981

0
...
9955
0
...
9975
0
...
9941
0
...
9967
0
...
9982

0
...
9957
0
...
9977
0
...
9945
0
...
9969
0
...
9984

0
...
9960
0
...
9978
0
...
9948
0
...
9971
0
...
9985

0
...
9962
0
...
9979
0
...
9951
0
...
9973
0
...
9986

0
...
9964
0
...
9981
0
...
0
3
...
2
3
...
4

0
...
9990
0
...
9995
0
...
9987
0
...
9993
0
...
9997

0
...
9991
0
...
9995
0
...
9988
0
...
9994
0
...
9997

0
...
9992
0
...
9996
0
...
9989
0
...
9994
0
...
9997

0
...
9992
0
...
9996
0
...
9989
0
...
9995
0
...
9997

0
...
9993
0
...
9996
0
...
9990
0
...
9995
0
...
9998

3
...
6
3
...
8
3
...
9998
0
...
9999
0
...
0000

0
...
9998
0
...
9999
1
...
9998
0
...
9999
0
...
0000

0
...
9999
0
...
9999
1
...
9998
0
...
9999
0
...
0000

0
...
9999
0
...
9999
1
...
9998
0
...
9999
0
...
0000

0
...
9999
0
...
9999
1
...
9998
0
...
9999
0
...
0000

0
...
9999
0
...
9999
1
...
It is defined by
1 Γ( 1 ν + 1 )
2
2
P =√
νπ Γ( 1 ν)
2

x
−∞

1

(1 + t2 /ν)− 2 (ν+1) dt
...
(X ≤ x) = P
...
90 P=0
...
975

0
...
995

0
...
9995

1
2
3
4
5

3
...
886
1
...
533
1
...
314
2
...
353
2
...
015

12
...
303
3
...
776
2
...
821
6
...
541
3
...
365

63
...
925
5
...
604
4
...
302
22
...
215
7
...
894

636
...
598
12
...
610
6
...
440
1
...
397
1
...
372

1
...
895
1
...
833
1
...
447
2
...
306
2
...
228

3
...
998
2
...
821
2
...
707
3
...
355
3
...
169

5
...
785
4
...
297
4
...
959
5
...
041
4
...
587

11
12
13
14
15

1
...
356
1
...
345
1
...
796
1
...
771
1
...
753

2
...
179
2
...
145
2
...
718
2
...
650
2
...
602

3
...
055
3
...
977
2
...
025
3
...
852
3
...
733

4
...
318
4
...
140
4
...
337
1
...
330
1
...
325

1
...
740
1
...
729
1
...
120
2
...
101
2
...
086

2
...
567
2
...
539
2
...
921
2
...
878
2
...
845

3
...
646
3
...
579
3
...
015
3
...
922
3
...
850

24
30
40
50
60

1
...
310
1
...
299
1
...
711
1
...
684
1
...
671

2
...
042
2
...
009
2
...
492
2
...
423
2
...
390

2
...
750
2
...
678
2
...
467
3
...
307
3
...
232

3
...
646
3
...
496
3
...
292
1
...
286
1
...
664
1
...
653
1
...
990
1
...
972
1
...
374
2
...
345
2
...
639
2
...
601
2
...
195
3
...
131
3
...
416
3
...
340
3
...
It is defined by
x 1
1
1
P =
u 2 ν−1 e− 2 u du
...
(X ≤ x) = P
...

ν

P = 0
...
01

0
...
05

0
...
975

0
...
995

0
...
0
2
...
0
4
...
0
6
...
0
8
...
0
10
...
04 393
0
...
07172
0
...
4117
0
...
9893
1
...
735
2
...
03 157
0
...
1148
0
...
5543
0
...
239
1
...
088
2
...
03 982
0
...
2158
0
...
8312
1
...
690
2
...
700
3
...
00393
0
...
3518
0
...
145
1
...
167
2
...
325
3
...
841
5
...
815
9
...
070
12
...
067
15
...
919
18
...
024
7
...
348
11
...
832
14
...
013
17
...
023
20
...
635
9
...
345
13
...
086
16
...
475
20
...
666
23
...
879
10
...
838
14
...
750
18
...
278
21
...
589
25
...
828
13
...
266
18
...
515
22
...
322
26
...
877
29
...
0
12
...
0
14
...
0
16
...
0
18
...
0
20
...
603
3
...
565
4
...
601
5
...
697
6
...
844
7
...
053
3
...
107
4
...
229
5
...
408
7
...
633
8
...
816
4
...
009
5
...
262
6
...
564
8
...
907
9
...
575
5
...
892
6
...
261
7
...
672
9
...
117
10
...
675
21
...
362
23
...
996
26
...
587
28
...
144
31
...
920
23
...
736
26
...
488
28
...
191
31
...
852
34
...
725
26
...
688
29
...
578
32
...
409
34
...
191
37
...
757
28
...
819
31
...
801
34
...
718
37
...
582
39
...
264
32
...
528
36
...
697
39
...
790
42
...
820
45
...
0
22
...
0
24
...
0
26
...
0
28
...
0
30
...
034
8
...
260
9
...
520
11
...
808
12
...
121
13
...
897
9
...
196
10
...
524
12
...
879
13
...
256
14
...
283
10
...
689
12
...
120
13
...
573
15
...
047
16
...
591
12
...
091
13
...
611
15
...
151
16
...
708
18
...
671
33
...
172
36
...
652
38
...
113
41
...
557
43
...
479
36
...
076
39
...
646
41
...
195
44
...
722
46
...
932
40
...
638
42
...
314
45
...
963
48
...
588
50
...
401
42
...
181
45
...
928
48
...
645
50
...
336
53
...
797
48
...
728
51
...
620
54
...
476
56
...
301
59
...
0
50
...
0
70
...
0
90
...
0

20
...
991
35
...
275
51
...
196
67
...
164
29
...
485
45
...
540
61
...
065

24
...
357
40
...
758
57
...
647
74
...
509
34
...
188
51
...
391
69
...
929

55
...
505
79
...
531
101
...
145
124
...
342
71
...
298
95
...
629
118
...
561

63
...
154
88
...
425
112
...
116
135
...
766
79
...
952
104
...
321
128
...
169

73
...
661
99
...
317
124
...
208
149
...
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
...
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
...
675 × 10−27 kg

1
...
110 × 10−31 kg

6
...
055 × 10−34 J s

1
...
673 × 10−11 N m2 kg−2

5
...
742 × 10−16 J m2 s−1

8
...
022 ×1023 mol−1

8
...
292 ×10−11 m

9
...
297 ×10−3

1
...
96 ×108 m

3
...
976 ×1024 kg

6
...
461 ×1015 m

1
...
086 ×1016 m
3
...
2415 × 1018 electronvolts (eV)



---

Title: formulae
Description: in these notes there are so many formulae which is used in mathematics

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