PDFs Week2Lecture1
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Transcript of PDFs Week2Lecture1
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8/13/2019 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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8/13/2019 PDFs Week2Lecture1
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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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8/13/2019 PDFs Week2Lecture1
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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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8/13/2019 PDFs Week2Lecture1
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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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8/13/2019 PDFs Week2Lecture1
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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