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Cheat sheet on Complex Variables with guided explanation
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A complex number is written as z = x + iy where x and y are real numbers and i^2 = −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 ¯z is defined a ¯z = x − iy. A complex number may be plotted in the 2D x − y plane 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| =
x^2 + y^2
Note that |z|^2 = z z¯. From the geometry in the plane we see that
z = r(cos(θ) + i sin(θ)), r = |z|, cos(θ) = x/r, sin(θ) = y/r
To add, multiply or divide two complex numbers z 1 = x 1 + iy 1 and x 2 + iy 2 we use i^2 = −1 ,
z 1 + z 2 = (x 1 + iy 1 ) + (x 2 + iy 2 ) = (x 1 + x 2 ) + i(y 1 + y 2 ), z 1 z 2 = (x 1 + iy 1 )(x 2 + iy 2 ) = (x 1 x 2 − y 1 y 2 ) + i(x 1 y 2 + y 1 x 2 ), z 1 z 2
= z^1 z¯^2 z 2 z¯ 2
= z^1 z¯^2 |z 2 |^2
= (x^1 +^ iy^1 )(x^2 −^ iy^2 ) |z 2 |^2
= (x^1 x^2 +^ y^1 y^2 ) +^ i(−x^1 y^2 +^ y^1 x^2 ) x^22 + y 22
From Taylor series
cos(x) = 1 − 1 2!
x^2 +^1 4!
x^4 +... , sin(x) = x − 1 3!
x^3 +^1 5!
x^5 +... ,
eix^ = 1 + ix +^1 2!
(ix)^2 +^1 3!
(ix)^3 +... =
x^2 +^1 4!
x^4 +...
x − 1 3!
x^3 +^1 5!
x^5 +...
This leads to Euler’s formula
eix^ = cos(x) + i sin(x)
and thus
z = r
cos(θ) + i sin(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) + i sin(y))
From geometry we have the triangle inequality for complex numbers ∣∣ |z 1 | − |z 2 |
≤ |z 1 + z 2 | ≤ |z 1 | + |z 2 |