06 - Complex functions
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welcome to week two in our course
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analysis of a complex kind this is the
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first lecture and it's on complex
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functions this week we will finally get
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to one of our big goals Julia sets of
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quadratic polynomials we'll also study
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the Mandelbrot set
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we basically laid the groundwork last
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week for these two topics but we need a
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little bit of more preparation we need
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to study complex functions and that's
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what this lecture is all about and we
00:30
need to study sequences and limits of
00:32
complex numbers and complex functions
00:35
that's what the next lecture is about
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after that we will be ready to study
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Julia sets and Mandelbrot set in
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particular we'll want to study functions
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and quadratic polynomials of this form f
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of Z equals Z squared plus C so let's
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review briefly would a function is a
00:55
function in general is sister rule a
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function f from A to B is a rule that
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assigns to each element of a exactly one
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element of B so for example the function
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f from R to R given by f of X equals x
01:12
squared plus 1 is a rule that takes a
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real number for example 2 and assigns to
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it the value 2 squared plus 1 which is 5
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or it takes the number minus 3 and
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assigns to it the value minus 3 squared
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plus 1 which equals 10 we often graph
01:38
functions in order to understand them
01:39
better so here you see the graph of the
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function x squared plus 1 so for example
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when X is equal to 0 then the function
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Maps 0 to 0 squared plus 1 which equals
01:53
1 and so when x is equal to 0 the Y
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value is equal to 1 which you see right
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here now imagine we want to study
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functions where we're not only plugging
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in real numbers into our functions but
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possibly complex numbers the one you
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graph worked for us for functions from
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the real number so the
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real numbers we had all the x-values on
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one axis and the corresponding y-values
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on the other axis and then we were able
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to plot this graph but if we're able to
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plug in not only real numbers but whole
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complex numbers the x-axis needs to
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become a plane namely the complex plane
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for the input values and it's going to
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be a little harder to graph such a
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function so a complex function is a
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function where we are allowing complex
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numbers to be plugged into the function
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and complex outputs to come out of the
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function so for example f of Z is equal
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to Z squared plus 1 not only do we allow
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X values here as inputs but for allowing
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complex numbers as inputs in the output
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is also a complex number but how could
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we graph this we would need four
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dimensions to graph this we would need a
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whole complex plane for the allowable
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arguments of the function and a whole
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complex claim for the allowable outcomes
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so let's study this a little bit further
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what does this function actually do
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typically when we look at complex
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numbers we look at their real part X and
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the imaginary part Y so let's write Z as
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X plus iy and let's find out what F of C
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equals f of C and we often call that WME
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to deal with complex numbers is then
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equal to x plus iy quantity squared this
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is our Z squared here this is just Z
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squared plus 1 and now that we wrote it
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in terms of x and y let's actually
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multiply out X plus iy quantity squared
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becomes x squared plus the second term I
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y1 these squared with I squared is minus
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1 that's where this negative sign comes
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from so minus y squared plus 2 times X
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times I why our that term back here 2xy
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times I and then there's this plus 1 and
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I sort it the terms that we get by
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multiplying out by real numbers and
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complex parts the real part of the
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result of multiplying out is x squared
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minus y squared plus 1 the imaginary
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part is 2xy
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we can let's write this function f as a
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real part that depends on x and y we
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call that U of XY and the imaginary part
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that also depends on x and y we call
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that V of x and y so in this example
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here we would have U of X and y equal x
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squared minus y squared plus 1 and V of
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XY would be equal to 2 times X times y
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so both these functions are functions
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that take inputs x and y and spit out
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something real and put together they
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form this complex number u plus IV so U
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and V are functions from we say R 2 to R
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because they have these two real inputs
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X and 1
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so how would you grab such a function
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the idea is to consider two complex
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planes one for the domain and that's
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right here so the domain is everything
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you allowed to plug into your function
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we call that the Z plane and one for the
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range so that's what the function spits
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out we call that the W plane and the
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function f maps the Z plane to the W
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plane in order for this to be useful
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we're going to want to analyze geometric
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configurations over in the Z plane and
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see what they get mapped to in the W
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plane so for example if we had something
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like this maybe we would want to analyze
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what does it get mapped to maybe it gets
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mapped to this over here in the W plane
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this will help us understand the
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function a little better so let's do
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that for the function we were just
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looking at so let's go back to F of Z
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equals Z squared and again W which is f
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of Z is then just x squared minus y
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squared plus 2i XY looking at it like
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this is still going to be difficult for
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graphing purposes so sometimes it is
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more useful to use polar coordinates in
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this example if we write Z as r eally i
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theta instead of writing zs x plus iy as
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we had done before these are just two
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different notations for the same thing
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but if we use polar notation then W
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which is Z squared so this number
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squared here becomes R squared times e
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to the I theta squared which is even the
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two I theta as we learned last week and
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so this formula for W the outcome tells
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us the length of W is given right here
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so that's where the length comes from
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and the argument of W you can find up
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here is two times the argument of Z and
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this really helps us in graphing the
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function or graphing geometric
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configurations and seeing what they get
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mapped to under this function again as a
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reminder W is f of Z which is Z squared
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we write Z
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as our leads the I theta so we had seen
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that the norm of W is the length of Z
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squared and an argument of W can be
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found by taking two times an argument of
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Z so what this function does it squares
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the distance from the origin of the
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number and it doubles the argument so if
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you look at a Z that moves in the Z
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plane around a circle of radius R then
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the corresponding WS would move around a
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circle of radius R squared at twice the
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speed for example this point right here
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the argument is zero would just get
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mapped over here but the argument is
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still zero but the distance on the
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origin has changed from R to R squared
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this point right here has an argument of
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PI over 2 doubling the argument will map
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this point over here again a distance R
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squared from distance R this point right
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here has an argument of PI if you double
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that argument it becomes 2 pi which is
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over here so it's going to agree with
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this point so pink and green are can on
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top of each other so both the pink point
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and the green point get mapped to this
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point over here this point is argument
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is 0 and distance from the origin is R
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squared so let's take a point in between
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how about this one right there that
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point let's say it has argument PI over
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4 if you double left the argument
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becomes PI over 2 and again it's going
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to sit on the circle of radius R squared
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let's take a point down here I'm going
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to run out of color so I'm going to use
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a little cross to distinguish it from
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this blue point right here the argument
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of this point down here is 3 PI over 2
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if I double that the argument becomes 6
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PI over 2 which equals 3 PI 2 pi will be
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growing around once the radii is over
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here so that blue point down here which
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I indicated with a cross
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SMAP to the same point that the orange
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point got mapped to namely over here
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let's look at a little movie in which I
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show you how a point moves around a
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circle of radius R here's another way of
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visualizing what this function does I
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graphed here a little sector and various
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radii that are indicated in different
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colors and also lines going out along
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radii the image of this can be seen over
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here so here is exactly what the
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function f would map this sector to if
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this is our radius R then this is
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becoming R squared right here so in this
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case actually R was 2 and so R squared
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is equal to 4 and we see that all
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arguments get doubled so for example
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this sector here gets opened up and
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becomes this size sector and so forth so
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this angle right there
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becomes a larger angle namely this one
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which is twice the previous angle or
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this angle right here becomes this angle
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right here and so forth so angle is get
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doubled and the radius is squared from 2
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to 4
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so how do you understand more
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complicated functions let's say the
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function wasn't just Z squared but it's
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Z squared plus 1 well the idea is the
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same except we can take this function
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apart into f of Z equals Z squared and a
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separate functional then just adds 1 to
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the result so we first look at a w1
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plane and just that the function w1 is
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equal to Z squared so we already
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understand how that function works and
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it indicated that here by drawing a
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little sector and its image under this
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function and then we compose that with
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the second function the function W
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equals w1 plus 1s and the W plane we
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just add one adding one shifts
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everything to the right because we're
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adding one not adding any imaginary
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parts we're just adding a real number 1
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so this 2 shifts this picture to the
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right and so this entire circle gets
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shifted to the right it is no longer
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centered at 0 but is now centered at 1
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and so all together if you just look at
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the function that goes all the way from
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here from the z plane to the W plane
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that will be the function f which is the
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function f of Z equals C squared plus 1
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you just took it apart into two steps
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and there I can understand it a little
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better so how about the function f of Z
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equals Z squared plus C these are the
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types of functions were interested in
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where C is now any complex number and
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not just the number one same idea we'll
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take it apart we'll first look at the
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function w1 is equal to Z squared which
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we just understood in the previous
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picture already and then the function W
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equals w1 plus C so we're adding this C
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together these two functions together
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executed one after the other
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keep the function W is equal to Z
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squared plus C and that's the function
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we're interested in now if you're adding
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C to a complex number that means you're
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shifting the circle that's so centered
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at 0 to a circle centered at C and
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that's what you can see right here when
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we talk about Julia
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sets the Mandelbrot set we talked about
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the iteration of functions so let's try
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to understand what that means we're
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interested in a function of the type f
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of f of z so we're plugging in f of z
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into the function f instead of just
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plugging in Z into the function f let's
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start by looking at an example suppose
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the f of Z is the function Z plus 1 all
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that function does is takes a point in
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the complex plane say this point right
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here and maps it to the right so it
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pushes it out further to the right so
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this was the original point here then it
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Maps it right there so it just moves it
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over by one to the right what happens if
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I then apply this function again where
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would then take this point if this was
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original point Z this is Z plus 1 and it
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would map that point and move it to the
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right one more time
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so it would go to Z plus 2 and then if I
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applied F again it would go to C plus 3
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let's actually plug in the number
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suppose Z is equal to I what's F of C F
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of C is simply I plus 1 in that case and
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what is if I apply f one more time what
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is f of f of Z in that case well that
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would be F of I plus one so I need to
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apply the function f to the number I
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plus 1 but F just adds 1 to whatever I
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stick in there so it's I plus 1 plus 1
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which is I plus 2 and then if I apply F
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one more time to that so I get F of F of
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F of C already calculated F of f of C
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that was I plus 2 and the function f
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here just adds 1 to whatever I see in
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there so it's going to be I plus 3 in
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general F of f of Z what is that F of Z
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it's
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alfe is c plus one and we're going to
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stick that back into the function f so
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into this function right here we're now
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sticking z plus one wherever you say z
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we're going to write z plus one instead
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so we're going to stick z plus one right
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there that gives us Z plus one quantity
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plus one or all together if we do the
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math it's Z plus two next we're going to
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take this one step further we're going
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to look at F of F of F of Z so that's
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the same thing as F of F 2 of Z where F
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2 of Z is too short for F of F of Z but
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F 2 of Z we just calculated we already
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know what that is F 2 of Z is Z plus 2
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so all we need to do is we need to look
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at F of Z plus 2 so now we need to stay
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Z plus 2 into the function f so wherever
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you see a Z we're going to write Z plus
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2 we're going to stick that into F so
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that gives us Z plus 2 plus 1 which is
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the plus 3 and you're starting to see
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your picture here f of Z is Z plus 1 F 2
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of Z is Z plus 2 a 3 of Z is Z plus 3 so
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you get the idea if you keep doing this
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if you do this n times M you get to F n
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of C and that's Z plus n F 2 is also
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written as f composed with F it's the
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composition of F with itself or F 3 is F
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composed with F composed with F and so
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forth so in other words F n would be F
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composed with F composed with F dot dot
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dot composed with F and I wrote F n
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times here we read this as F n sometimes
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people write F and they put a little
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circle up here before the end to
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indicate we're talking
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composition of functions so executing
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function one after the other this is not
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the nth power of f the nth power of F is
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something entirely different
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let me demonstrate to you what the nth
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power of this F is just as a comparison
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the nth power of F as a comparison is
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the following it is f of Z you calculate
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F of Z and then raise the result to the
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nth power so notice the difference in
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notation here we have F n of zeod here
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we have F of Z to the power n admittedly
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this notation is confusing and it could
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be easily confused with the nth power
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but for now F n FC means the nth iterate
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so we just have to understand and be
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very clear about what we mean with our
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notation if we were interested in the
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nth power of F this would be F of C is C
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plus 1 and then taking that to the nth
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power and that is something like you
19:08
know if you use a binomial theorem this
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is C to the N plus n times e to the N
19:15
minus 1 and so forth you know higher
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this is a complicated formula and not at
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all the same thing as the nth iterate of
19:24
that so we're talking about the end
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iterative F
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so stop with another example say f of Z
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equals three Z what is the second
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iterate of F F 2 of Z is f of f of Z so
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we're plugging f of Z into the function
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f so wherever you see Z we're going to
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write F of Z instead so it's F of 3 Z
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because F of Z is 3 Z so we're plugging
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in 3 Z into this function f so it's 3
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times 3 times Z which is 3 squared times
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C what is F 3 of Z F 3 of Z is f of F 2
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of Z as we saw before so now we're going
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to plug F 2 of Z into that but we
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already know what F 2 of Z equals we
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calculated that's three square times Z
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so now we're plugging 3 squared times Z
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into the function f plugging in 3
20:31
squared times Z into the function f
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gives you 3 times 3 squared Z which is 3
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cubed Z all together so we see F of Z is
20:41
3 Z F 2 of Z is 3 squared Z if 3 of Z is
20:45
3 cubed Z and you get the idea that F n
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of Z is most likely 3 to the n times see
20:53
two more examples let's look at F of Z
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equals Z to the D F 2 of Z is then F of
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f of Z so that equals F of Z to the D
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which means we need to plug in Z to the
21:16
D into the function where you see Z
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we're going to replace that with Z to
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the T so it's Z to the D that's just
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this part right here and we need to
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raise that to the power T but if you
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raise 1 power to another power you can
21:32
multiply the two exponents with each
21:34
other so this is Z to the d squared I'll
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put parentheses around there to make
21:39
that very clear and again the end
21:42
that'll rid of these then obtained by
21:44
taking Z to the D to the N last example
21:49
let's look at F of Z equals Z squared
21:53
plus two so this is exactly a function
21:56
of the form that we're interested in we
21:58
want to study functions of the form f of
22:00
Z equals Z squared plus C so Z squared
22:06
plus 2 that's the function that we
22:08
obtain when C is equal to 2 so what is f
22:11
2 of Z that is F of F of Z so what is f
22:19
of Z its Z squared plus 2 so we need to
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plug in Z squared plus 2 into the
22:25
function f so wherever you see a Z we're
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going to replace that with Z squared
22:28
plus 2 so we get Z squared plus 2 that's
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the quantity that we're going to square
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and then we add 2 to that and that's
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what you see right here when you
22:41
multiply this through Z squared plus 2
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quantity squared is Z to the fourth plus
22:47
2 times 2 times Z squared so 4 times Z
22:49
squared plus 4 and then this plus 2 here
22:52
gives you this plus 6 F 3 of Z takes
22:56
that number squares it and adds another
22:58
2 to it
22:59
so it takes the Z to the fourth plus 4 z
23:02
squared plus 6 squares it and that's
23:05
another 2 to it when you multiply this
23:07
through you'll see that there's a Z to
23:09
the 8th thurman and lots of other terms
23:11
but the highest power of Z is going to
23:13
be Z to the 8th and in general F n is a
23:17
polynomial of degree 2 to the nth so
23:20
it's going to start with F n of Z is
23:25
equal to Z to the 2 to the N and then
23:31
lower powers of C it's going to be a
23:33
polynomial but it's a of a high degree
23:35
so this gets pretty complicated pretty
23:38
quickly looking at the end little bit of
23:40
a rather simple function
23:47
when we studied Julia sets of quadratic
23:51
polynomials we studied quadratic
23:54
polynomials of the form Z squared plus C
23:56
and we'll see why we don't have to study
23:59
other quadratic polynomials in general
24:02
if you look at quadratic functions you
24:04
could have something like 3z squared
24:06
minus 15 z plus 4 that's not exactly in
24:12
this form Z squared plus C right Z
24:15
squared plus C doesn't allow for a
24:16
linear term and it doesn't allow for any
24:18
number here in front but we'll see once
24:21
we start studying Julia sets Y it is
24:23
enough to just look at these types of
24:26
polynomials and understand none and
24:28
therefore we'll be able to understand
24:30
more general ones as well and we will
24:33
study the behavior of the iterates of
24:36
this functions F f2 f3 f4 FN the Julia
24:42
set of such a polynomial is going to be
24:44
the set of all those points Z that we
24:47
can plug in for which the sequence of
24:50
iterates behaves chaotically
24:52
we'll have to define exactly what we
24:55
mean by that but it'll become pretty
24:57
clear by looking at some examples we
25:01
need that's a little bit more
25:02
preparation we need to study sequences
25:05
of complex numbers so for example one Z
25:09
is equal to zero we would study the
25:14
sequence 0 F of 0 F of F of 0 F of F of
25:27
F of 0 and so forth
25:31
so what sequence is that let's actually
25:32
do the math here what is f of 0 if you
25:35
plug in 0 into this function you just
25:37
get C 0 0 C F of F of 0 so now we're
25:43
plugging F of 0 which is C into this
25:45
function f of C is C squared plus C so
25:49
we'll get a C squared plus C F of that
25:53
whole thing now apply on c squared plus
25:55
c back into the function f so that's
25:58
going to be C squared plus C
26:00
want to be squared and then we're going
26:02
to add another seat to it and it keeps
26:05
going like that so you could multiply
26:09
through and you would get that this is a
26:12
sequence of complex numbers and the
26:14
question we're going to ask is what does
26:16
that sequence do as we keep going does
26:19
it run around wildly in the complex
26:21
plane is it going circles Ithaca often
26:24
finiti does it come to 0 does it get
26:26
closer and closer to this one point what
26:28
does the sequence do so we need to study
26:31
a little bit more what are sequences of
26:34
complex numbers and that's what we'll
26:36
study next