Complex Variables Cheat Sheet
A complex number is written as z=x+iy where xand yare real numbers and i2=−1. We
write <(z) = x(or Re(z)) for the real part of z and =(z) = y(or Im(z)) for the imaginary part of
z.The complex conjugate, denoted by z∗or ¯zis defined a ¯z=x−iy. A complex number may
be plotted in the 2D x−yplane known as the complex (or Argand) plane. It can be viewed as a
vector or point in 2D with coordinates (x, y). The length or modulus of z=x+iy is the length of
the vector,
|z|=px2+y2
Note that |z|2=z¯z. From the geometry in the plane we see that
z=r(cos(θ) + isin(θ)),
r=|z|,cos(θ) = x/r, sin(θ) = y/r
To add, multiply or divide two complex numbers z1=x1+iy1and x2+iy2we use i2=−1 ,
z1+z2= (x1+iy1)+(x2+iy2)=(x1+x2) + i(y1+y2),
z1z2= (x1+iy1)(x2+iy2)=(x1x2−y1y2) + i(x1y2+y1x2),
z1
z2
=z1¯z2
z2¯z2
=z1¯z2
|z2|2=(x1+iy1)(x2−iy2)
|z2|2=(x1x2+y1y2) + i(−x1y2+y1x2)
x2
2+y2
2
From Taylor series
cos(x) = 1 −1
2!x2+1
4!x4+. . . , sin(x) = x−1
3!x3+1
5!x5+. . . ,
eix = 1 + ix +1
2!(ix)2+1
3!(ix)3+. . . =1−1
2!x2+1
4!x4+. . . +ix−1
3!x3+1
5!x5+. . .
This leads to Euler’s formula
eix = cos(x) + isin(x)
and thus
z=rcos(θ) + isin(θ)=r eiθ (polar form),
r=|z|,cos(θ) = x/r, sin(θ) = y/r
We define ez, cos(z), sin(z) etc. from the Taylor series. We have
ez=ex+iy =exeiy =ex(cos(y) + isin(y))
From geometry we have the triangle inequality for complex numbers
|z1|−|z2|
≤ |z1+z2| ≤ |z1|+|z2|
1