Mathematical Formulae You Should Know

Exponentials and Logarithms

eln x = x, ln(e) = 1, ln(ea) = a, ln(xa) = a ln x.

eaeb = ea+b, ln(xy) = lnx + ln y, ln(x/y) = lnx − ln y

Complex numbers

The imaginary number i is defined by i2 = −1; j is sometimes used instead. A complexnumber can be written in terms of a real and imaginary part, or in terms of a modulusr and an angle θ:

z = x + iy = r e+iθ = r [cos θ + i sin θ].

Under complex conjugation i → −i:

z∗ = x − iy = r e−iθ = r [cos θ − i sin θ].

The modulus squared of z is the real number |z|2 = z∗z = x2 + y2 = r2. Note also

cos θ =1

2[eiθ + e−iθ], sin θ =

1

2i[eiθ − e−iθ].

Trigonometry

sin2x + cos2x = 1

sin(x ± y) = sin x cos y ± cos x sin y

cos(x ± y) = cos x cos y ∓ sinx sin y

The following may be derived easily from the above:

sinx cos y = 1

2

(

sin(x + y) + sin(x − y))

cos x cos y = 1

2

(

cos(x + y) + cos(x − y))

sinx sin y = 1

2

(

cos(x − y) − cos(x + y))

sinx ± sin y = 2 sin 1

2(x ± y) cos 1

2(x ∓ y)

cos x + cos y = 2cos 1

2(x + y) cos 1

2(x − y)

cos x − cos y = −2 sin 1

2(x + y) sin 1

2(x − y)

sin 2x = 2 sin x cos x

cos 2x = cos2 x − sin2 x = 2cos2 x − 1 = 1 − 2 sin2 x

Hyperbolic functions

cosh x = 1

2[ex + e−x], sinhx = 1

2[ex − e−x]

Power Series

Geometric series:

1 + x + x2 + . . . + xn−1 =1 − xn

1 − x

n→∞−→ (1 − x)−1 −1 < x < 1

Binomial expansion:

(1 + x)n = 1 + nx +n(n − 1)

2!x2 +

n(n − 1)(n − 2)

3!x3 + . . .

If n is a positive integer the series terminates after n+1 terms, otherwise it is an infiniteseries.

Taylor Series about the point x = x0:

f(x) = f(x0) + (x − x0)df

dx

∣

∣

∣

∣

x0

+1

2!(x − x0)

2 d2f

dx2

∣

∣

∣

∣

x0

+ . . .

The Maclaurin series is the Taylor series about the point x = 0. The following Maclaurinseries are frequently encountered:

ex = 1 + x +x2

2!+

x3

3!+ . . . all x

cos x = 1 − x2

2!+

x4

4!+ . . . all x

sinx = x − x3

3!+

x5

5!− . . . all x

ln(1 ± x) = ±x − x2

2± x3

3− . . . −1 < x < 1

(1 ∓ x)−1 = 1 ± x + x2 ± x3 + . . . −1 < x < 1

L’Hopital’s rule follows from Maclaurin series: if f(0) = g(0) = 0,

limx→0

f(x)

g(x)= lim

x→0

df/dx

dg/dxeg. lim

x→0

sinx

x= 1.

Differentiation and Integration

f(x) df/dx

xn nxn−1

eax aeax

lnx 1/x

sinx cos x

cos x − sin x

sinhx cosh x

cosh x sinhx

f(x) df/dx

arcsin(xa)

1√a2 − x2

arccos(xa) − 1√

a2 − x2

arctan(xa)

a

a2 + x2

arcsinh(xa)

1√a2 + x2

arccosh(xa)

1√x2 − a2

In the above a is a constant.This table works in reverse to give a table of common integrals as well, except that∫

(1/x) dx = ln |x|+ c. Remember to add the constant of integration.

Product rule and integration by parts:

d

dx[f(x)g(x)] = f(x)

dg

dx+ g(x)

df

dx⇒

∫

f(x)dg

dxdx = f(x)g(x)−

∫

g(x)df

dxdx + c

Chain rule

d

dxf(

g(x))

=df

dg· dg

dx

Partial Differentiation

Given a function f of more than one variable, the infinitesimal change in f when thevariables are changed infinitesimally is denoted df . For f = f(x, y),

df =

(

∂f

∂x

)

y

dx +

(

∂f

∂y

)

x

dy

If in turn x and y are functions of other variables, say r and φ, the chain rule gives

(

∂f

∂r

)

φ

=

(

∂f

∂x

)

y

(

∂x

∂r

)

φ

+

(

∂f

∂y

)

x

(

∂y

∂r

)

φ

Differential Equations

Important differential equations and their solutions:

y − λy = 0

y + ω2y = 0

y + γy + ω20y = 0

y = Aeλt

y = A cos(ωt + φ) or y = B cos ωt + C sin ωt

y = Ae−1

2γt cos(ωt + φ)

where A, B, C and φ are constants, and in the last equation ω =√

ω20 − 1

4γ2 and the

solution given is correct for the underdamped case, 2ω0 > |γ|.

Vectors

A vector can be expressed in terms of 3 orthogonal, normalised basis vectors which arecommonly denoted i, j and k (or x, y and z): a = axi + ayj + azk, or represented bythe number triplet of its components: a = (ax, ay , az).

The modulus of a is |a| =√

a2x + a2

y + a2z =

√a · a

Scalar product: if θ is the angle between two vectors a and b, the scalar product is

a · b = axbx + ayby + azbz = |a||b| cosθ

Vector product:

a× b = (aybz − azby)i + (azbx − axbz)j + (axby − aybx)k = |a||b| sinθ n

where n is a unit vector perpendicular to a and b in a direction such that a, b and n

form a right-handed set.

Scalar triple product: a · (b× c) = b · (c × a) = c · (a× b)

Vector triple product: a × (b× c) = b(a · c) − c(a · b)

Cartesian and Polar Coordinate Systems

The position vector is usually called r: r = xi+yj+zk, where x, y, and z are the cartesiancoordinates of the point. The radial distance from the origin is r = |r| =

√

x2 + y2 + z2

and it is frequently useful to define the unit vector r = r/r.

Plane polar coordinates: (r,φ) or (r, θ)

r =√

x2 + y2; φ = arctan(y/x);x = r cos φ; y = r sinφ;area element: dA = r dr dφ

Cylindrical polar coordinates: (r,φ,z)Coordinates defined as in plane polars, with the addition of the third coordinate z;volume element: dV = r dr dφ dz

Spherical polar coordinates: (r,θ,φ)

r =√

x2 + y2 + z2; θ = arctan(√

x2 + y2/z); φ = arctan(y/x);x = r sin θ cos φ; y = r sin θ sinφ; z = r cos θvolume element: dV = r2 sin θ dr dθ dφ;area element on the surface of the sphere: dA = r2 sin θ dθ dφ r

Vector Caculus

Grad(φ) ≡ ∇φ =∂φ

∂xi +

∂φ

∂yj +

∂φ

∂zk

Div(a) ≡ ∇ · a =∂ax

∂x+

∂ay

∂y+

∂az

∂z

Curl(a) ≡ ∇× a =

(

∂az

∂y− ∂ay

∂z

)

i +

(

∂ax

∂z− ∂az

∂x

)

j +

(

∂ay

∂x− ∂ax

∂y

)

k

∇2φ =∂2φ

∂x2+

∂2φ

∂y2+

∂2φ

∂z2