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Complex Variables Cheat Sheet, Cheat Sheet of Complex analysis

Cheat sheet on Complex Variables with guided explanation

Typology: Cheat Sheet

2019/2020

Uploaded on 11/27/2020

myboy
myboy 🇺🇸

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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 zor ¯zis defined a ¯z=xiy. A complex number may
be plotted in the 2D xyplane 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)=(x1x2y1y2) + i(x1y2+y1x2),
z1
z2
=z1¯z2
z2¯z2
=z1¯z2
|z2|2=(x1+iy1)(x2iy2)
|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) = x1
3!x3+1
5!x5+. . . ,
eix = 1 + ix +1
2!(ix)2+1
3!(ix)3+. . . =11
2!x2+1
4!x4+. . . +ix1
3!x3+1
5!x5+. . .
This leads to Euler’s formula
eix = cos(x) + isin(x)
and thus
z=rcos(θ) + isin(θ)=r e (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

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Complex Variables Cheat Sheet

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 +...

  • i

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 |