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Reference Text for Mathematical Physics
Deepanshu Bisht

Contents
1 Some Common Standard Math Formulas
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1
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1
1
2
3

2 Some less used formulae
2
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2
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5
5
5

3 Partial Derivatives
3
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3
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6
6
6

4 Differential Equations
7
4
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7
4
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10
4
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10
5 Complex Analysis
11
5
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11
5
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12
5
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14
6 Vector Calculus
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16
7 Coordinate Systems
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8 Binomial theorem, Permutation and Combinations

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22

9 Matrices and Determinants
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24
10 Limits and Series
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27
11 Standard Functions
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1

28
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29
32

12 Everything Fourier
33
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33
12
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34
13 Stats and Probability
35
13
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37
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38

2

About the document
This document is a go-to reference in need of mathematical formulae/equations and associated notes
which are directly or indirectly useful in doing physics
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Chapter 1

Some Common Standard Math
Formulas
1
...
1
...
1)
(1
...
3)

sin(a + b) = sin(a) cos(b) + cos(a)sin(b)

(1
...
5)

cos(a + b) = cos(a) cos(b) − sin(a) sin(b)

(1
...
7)

tan(a + b) =

tan(a) + tan(b)
1 − tan(a) tan(b)

(1
...
9)

sin(2a) = 2 sin(a) cos(a)

(1
...
11)
(1
...
13)
(1
...
15)
(1
...
17)
(1
...
1
...
2

(1
...
20)
(1
...
22)
(1
...
24)

Derivatives
d
(sin x) = cos x
dx
d
(cos x) = − sin x
dx
d
(tan x) = sec2 x
dx
d
(cot x) = −cosec2 x
dx
d
(sec x) = sec x tan x
dx

(1
...
26)
(1
...
28)
(1
...
30)

d
1
(sin−1 x) = √
dx
1 − x2

(1
...
32)

d
1
(tan−1 x) =
dx
1 + x2
d
−1
(cot−1 x) =
dx
1 + x2
1
d
(sec−1 x) = √
dx
x x2 − 1
d
−1
(cosec−1 x) = √
dx
x x2 − 1
d x
(e ) = ex
dx
d
1
(log|x|) =
dx
x
 x 
d
a
= ax
dx loga

2

(1
...
34)
(1
...
36)
(1
...
38)
(1
...
3

Integrals

Note: Evaluating the integral by changing the variables through substitution is all well and good
until
...
That is, x = f −1 (t) is a multi-valued function
...
40)
0

If for this integral we make the substitution sin x = t, both upper and lower limits become zero and
integral turns out to be 0
...
This happens cause x = sin−1 (t)
is multivalued
...

For the substitution cos x = t this doesnt happen since cos x doesnt repeat in 0 ≤ x ≤ π

1
...
1

Algebraic

ˆ
xn dx =

xn+1
+c
n+1

(1
...
3
...
42)

px + q
A
B
=
+
(x − a)2
x − a (x − a)2

(1
...
44)

px2 + qx + r
A
Bx + C
=
+ 2
2
(x − a)(x + bx + c)
x − a x + bx + c

(1
...
46)

ln|x|dx = xln|x| − x + c

(1
...
48)
(1
...
50)
(1
...
52)
(1
...
3
...
54)

cot x dx = ln| sin x| + c

(1
...
56)

cosec x dx = ln|cosec x − cot x| + c

(1
...
58)

cot2 xdx = −cot x − x + c

(1
...
60)
(1
...
62)
(1
...
64)
(1
...
66)
(1
...
3
...
3
...
68)

−αx2

dx =

(1
...
70)
(1
...
72)

Chapter 2

Some less used formulae
2
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2
...
1

Inverse trigonometry
cos−1 x + cos−1 x = π [−1 ≤ x ≤ 1]
sec

−1
−1

x + sec

−1

x = π |x| ≥ 1

−1

x=π x∈R
π
sin−1 x + cos−1 x = [−1 ≤ x ≤ 1]
2
π
tan−1 x + cot−1 x =
x∈R
2
π
cosec−1 x + sec−1 x = |x| ≥ 1
2
cot

2
...
1)
(2
...
3)
(2
...
5)
(2
...
7)

(2
...
1

Basics

Defined as:

∂f
f (x + ∆x, y) − f (x, y)
=
∂x
∆x
∂f
f (x, y + ∆y) − f (x, y)
=
∂y
∆y

(3
...
2)

for function f (x, y)
The total differential is given as: measures the total change in f , due to changes in x and y
df =

∂f
∂f
dx +
dy
∂x
∂y

(3
...

(3
...
Further for such an f ,
fxy = fyx so we get a condition for exactness
∂B
∂A
=
∂y
∂x

df = A(x, y)dx + B(x, y)dy

(3
...
2

Useful theorems




3
...
1

∂x
∂y

∂x
∂y
 
z




=
z

∂y
∂z

 
x

∂y
∂x

−1

∂z
∂x



(3
...
7)

y

Stationary points

The points of two variable functions f (x, y) at which the first partial derivatives become 0
...
1

Ordinary Differential Equations

4
...
1

Equations reducible to separable
dy
= f (ax + by + c)
dx
 
dy
y
=f
dx
x

(4
...
2)

Homogenous
f1 (x, y)dx + f2 (x, y)dy = 0

(4
...
4)

Change the variables: x = X + h, y = Y + k so equation becomes homogenous

4
...
2

Solution of exact equations

The equation is:
M dx + N dy = 0

(4
...
6)

(4
...
8)

(4
...
10)

• If ODE is such that:
f1 ydx + f2 xdy = 0
then
IF =

1
Mx − Ny

(4
...
12)

Linear first order equations
dy
+ P (x)y = Q(x)
dx
´

IF = e

(4
...
1
...
13)

Q(x)IF dx = C

(4
...
16)

where α is real
Put u(x) = y 1−α and the equation becomes linear in u
u0
+ p(x)u = q(x)
1−α

4
...
4

(4
...
18)

Find λ1 and λ2 : the roots of the auxiliary equation:
λ2 + aλ + b = 0

(4
...
20)

y(x) = (c1 + c2 x)eλx

(4
...
22)

Real and distinct

Real and equal

Complex

Then we have the following particular solutions
y1
y2

=

˙ 1 )x
e(a1 +ιb

(4
...
24)

=

ea1 x [cos(b1 x) − ι˙ sin(b1 x)]

(4
...
Subtract both and divide by 2˙ι
...
Their
general linear combination is general solution
y(x) = ea1 x [A cos(b1 x) + B sin(b1 x)]

4
...
5

(4
...
27)

m2 + (a − 1)m + b = 0

(4
...
29)

y(x) = (c1 + c2 ln x)xm

(4
...
31)

Real and equal

Complex

4
...
6

Linear second order Non homogeneous
y 00 + p(x)y 0 + q(x)y = r(x)

(4
...
33)

y 00 + ay 0 + by = r(x)

(4
...
If r(x) is sum of forms given in table, then assume yp also as sum of the corresponding yp
forms
Method of variation of parameters
If y1 and y2 are particular LI solutions of homgoneous solution yh then yp is:
ˆ
ˆ
y1 r(x)
y2 r(x)
dx + y2
dx
yp = −y1
W
W
Where W = y1 y20 − y2 y10 is wronskian

9

(4
...
2

Linear Systems of ODEs

The general form of such a system of ODEs is:
y10

=

a11 y1 + a12 y2 + · · · + a1n yn + g1 (t)

(4
...


...


...
37)

yn0

an1 y1 + an2 y2 + · · · + ann yn + gn (t)

(4
...
38)

There are n unknown variables and n linear ODEs
...
So in matrix form:
y0 = Ay + g
(4
...
2
...
41)

(4
...
43)

(4
...
3
4
...
1

(4
...
46)

(4
...
48)

Parabolic
Here we take ξ(x, y) = Constant and η = y which gives:
C ∗ uηη + D∗ uξ + E ∗ uη + F ∗ u = G∗

(4
...
50)

Elliptic

where, α =

ξ+η
2

and β =

ξ−η
2i

10

Chapter 5

Complex Analysis
Somehow, pretending that square roots of negative numbers made sense, even though they obviously did
not, could lead to sensible answers
...
But Bombellis calculation implied that there was more to
imaginaries than that
...

˙
eιπ
= −1

(5
...
But this equation should
not be interpreted as have we raised e to this weird power that doesn’t make sense
...
1

Basics

• Complex numbers are the most general set of numbers and include every other set of numbers
...

Both are equ

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