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Contents

Chapter 7: Complex Numbers

I. Definitions and RepresentationII. PropertiesIII. Differentiaton of complex functions

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I. Definitions and Representations

I. Definitions and Representations

A. General Definition:• iy is called an imaginary number (y R)

• Square cube, and 4th power etc…

• C numbers are: z=x+iy

• Complex Conjugate:

E7.1-1 Using the fundamental definition of i (=) compute the roots of the polynomial x2 +2x+2=0

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I. Definitions and Representations

I. Definitions and Representations

B. Operations:• Addition• Multiplication:

• Zero element (w/ respect to +)

• Identity element (with respect to -)

• Inverse and division

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I. Definitions and Representations

E7.1-2 Compute z-1 for z=3i ; z=1-2i; 1+2i . Write answer in the form z=a+ib where a and b are real.E7.1-3 Compute the magnitude of the following complex numbers by computing directly zz*: z=i z=1+i z= 1/(2-3i) z=(1+i)/(1-i)

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I. Definitions and Representations

I. Definitions and Representations

C. Argand (or complex) plane representation:

E7.1-4 What transformation, in the complex plane, does the multiplication by i induce? (prove statement)

Interpretation of multiplication z=z1z2 as a transformation in Argand plane:

E7.1-5 Find r and q for z=1+i; z=1; z=i; z=-2-3i, and z=-1 and write these numbers in angle magnitude (Argand variables) form: z=r (cosq +isin )q .

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I. Definitions and Representations

I. Definitions and Representations

D. Exponential representation:

• Taylor series: +…

• MacLaurin Series expansion: +… or +…

• Series for Cos: =…• Series for Sin: +…

• Series for Cos +i Sin : +…

• Series for exp: and thus +…. Which is the same as above and allows us to write the following:

E7.1-6 Write; z=i, z=-1, z=-1-2i and z1=1+i, z2=1/(1+i) in exponential form then compute z1*z2

and show that you get the expected results.

General expression for z in exponential notation:

Note how multiplication becomes easy:

( s ) iz Co iSin e

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II. Fundamental Properties

II. Fundamental Properties

Roots of polynomials:

Quadratic polynomials roots

Roots of unity: zn=1 and nth root of a complex number

E7.2-1 Find the cubic roots of unity and plot the solutions or the Argand plane

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III. Differentiation

III. Differentiation

The derivative of f is well defined if the above limit exists and is unique.Notice that since is complex there are an infinite number of ways z can approach zero and thus we are lead to strict constraints on f to be differentiable.

0

( ) ( ) ( )lim

df z f z f z

dz

Let’s compute for a function f:

However since z lives in the Argand plane there are only 2 independent ways to approach zero: along the real axis, or along the imaginary axis. Forcing these 2 paths to lead to the same limit imposes the so called Cauchy Riemann equations on f, leading us to a sufficient condition for f to be differentiable. First, for convenience we write:

First let’s compute the limit when z approaches 0 along the real axis: z = e +i0

0

{ ( , ) ( , y)} { ( , ) (x, y)}lim

u x y iv x u x y iv u vi

x x

Then let’s compute the limit when z approaches 0 along the imaginary axis: z = 0+id

0

{ ( , ) ( , y )} { ( , ) (x, y)} 1lim

u x y iv x u x y iv u vi

i i y y

( , *) ( , ) ( , ) ( , )f z z f x y u x y iv x y

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III. Differentiation

III. Differentiation

In order for the derivative to be meaningful we thus have to have these 2 limits equal. This will guaranty that no matter how z approaches zero (ie along what direction in the complex plane) the limit will be unique and thus the derivative will be well defined and unique . Equating the real parts of the previous equations as well as the imaginary parts, we get the following set of 2 conditions, called Cauchy-Riemann equations, for a function to be differentiable with respect to the complex variable z=x+iy:

Given:

and defining f in terms of its real part u(x,y) and imaginary part v(x,y)

( , *) ( , ) ( , )

Where u and v are real functions, we get the Cauchy-Riemannn conditions on f to be differe

z x iy

f z z u x y iv x y

ntiable:

u v

x y

v u

x y

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III. Differentiation

III. Differentiation

E7.3-1 Use the Cauchy-Riemann equations to find the derivatives, with respect to z, of the following functions (careful since you cannot assume that z is real) (find u(x,y) and v(x,y): A. B. C. Compare your answers to what you would have obtained if z was real.

E7.3-2 Use the fact that we have defined what is, in order to define and in terms of and compute the derivatives. Again compare your answers to what you would have gotten if z was real.

E7.3-3 Compute the derivative of z2 and the derivative of zz* and z*2 from first principles (using limits): use first an approach along x then along y. Comment on uniqueness and thus existence in both cases.

E7.3-4 Now consider the derivative of z*2 with respect to z* paralleling the approach we used for z. In general, is a function f(z, z*) differentiable with respect to either z or z*?

C. Properties of derivatives: once we have defined differentiation, it is easy to prove the usual properties of derivatives, including chain rule and product rule:

and {f(z). g(z)}’= f’(z). g(z)+ f(z). g’(z)