PDFs Week2Lecture1

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Analysis of a Complex KindWeek 2

Lecture 1: Complex Functions

Petra Bonfert-Taylor

Lecture 1: Complex Functions Analysis of a Complex Kind


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!"#$%&" (% !"") (*This week:

Julia sets for quadratic polynomials. The Mandelbrot set. We laid the ground last week! Just a little more preparation:

o Complex functions (Lecture 1).o Sequences and limits (Lecture 2).

Well need to study quadratic polynomials oform .f(z) =z2 +c


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-./$01%/2

Recall:A function is a rule that aseach element of exactly one element of

Example: The graph helpsus understand

the function.

f :

A

BA

f : R R, f(x) =x2 + 1y


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3%&4#"5 -./$01%/2

Now: How do we graph this? Need 4 dimensions? Writing we see:

f :C C

, f(z) =z2

+ 1

z = x+ y

w=f(z) = (x+iy)2 + 1

= (x2 y2 + 1) +i

= u(x, y) +iv(x, y)

where u, v: R2


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>58&4#"f(z) =z2, so w= (x+iy)2 = (x2

y2) +

?%0 2% .2"=.#@

A%7" .2"=.# 1/ 0912 $82"B 4%#87 $%%7C1/80"2+

z = rei, then w = r

2e2i

, so

w|=|z|2 and arg w = 2 arg z.


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As z moves around a circle of radius r once,moves around the circle of radius r2at doub

speed, twice.

w=f(z) =z2, so w=r2e2i

;


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w=f(z) =z

&%G1" 9"7" 8H%.0 7%0801/: 0*1$"

2


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w=f(z) =z2


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How do we understand more complicatedfunctions, such as ?

Same idea!

A%7" 3%&4#1$80"C -./$0

f(z) =z2 + 1

; *I **

I

D ;J * D *I

K I


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And how about ? Same idea (again)!

L%* 8H%.0MMM

; *I **ID ;

J * D *IK $

f(z) =z2 +c, c C

* D ;JK $


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N0"7801%/ %= -./$01%/Let f(z) = z+1. Then

f2(z) = f(f(z)) = f(z+1) = (z+1)+ 1 = z+2. f3(z) = f(f2(z)) = f(z+2) = (z+2)+ 1 = z+3. ... fn(z) = z+n. fn(read: Eff n) is called the nth iterate of f.

be confused with the nth power of f.)


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O/%09"7 >58&4#"Let f(z) = 3z. Then:

f2(z) = f(f(z)) = f(3z) = 3*3z = 32z. f3(z) = f(f2(z)) = 3*32z= 33z. ... fn(z) = 3nz


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(*% A%7"MMMLet f(z) = zd. Then:

f2(z) = (zd)d= z(d ) ... fn(z) = z(d )Now let f(z)= z

2

+ 2. Then: f2(z) = (z2 + 2)2+ 2 = z4+ 4z2+ 6 f3(z) = (z4+ 4z2+ 6)2+ 2 = z8+ ... fnis a polynomial of degree 2n.

!

"


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P.#18 Q"02 To study the Julia set of the polynomial f(z) =

well study the behavior of the iterates f, f2, f

fn, ... of this function.

The Julia set of f is the set of points z in the cplane at which this sequence of iterates be

chaotically.

We thus need one more preparation: We nestudy sequences of complex numbers. That

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