11 - The complex derivative

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welcome to our third week in our course

00:03

analysis of a complex kind today we will

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learn about the complex derivatives

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before we go there let's quickly review

00:12

the derivative from calculus giving a

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real valued function f of a real

00:18

variable so there's nothing complex in

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here and a point X 0 in the interval in

00:25

which F is defined we say the function

00:26

is differentiable at X 0 if the limit of

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the difference quotient exists and the

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difference quotient is f of X minus f of

00:36

X 0 divided by X minus X 0 if that limit

00:40

as X goes to X 0 exists we call the

00:43

limit the derivative of F at X 0 and

00:45

denoted by F prime of X 0 let's quickly

00:49

review what this means geometrically

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here's a graph of a function in this

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blue graph here is the graph of F in

00:59

this example now we pick a point X 0

01:04

where F is defined and we look at f of X

01:08

0 that's this point right here so this

01:10

point that I'm circling has coordinates

01:13

X 0 and f of X 0 and then we take a

01:16

second to point X and we look at f of X

01:20

and so this point that I'm now circling

01:22

as the point is coordinates are x and f

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of X next we look at the numerator of

01:30

our difference quotient f of X minus f

01:34

of X 0 f of X is this value right here f

01:37

of X 0 is this value the difference

01:39

between the two of them is this length

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so this is f of X minus f of X 0 and we

01:45

divide that by X minus X 0 X minus X 0

01:50

you can see in this length right there

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so this thing up here f of X minus f of

01:55

X 0 divided by X minus X 0 is rise over

01:59

run for this red secant line that

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actually in other words it gives us the

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slope of the red secant line is the

02:08

change in Y divided by the change in X

02:10

so this difference quotient f of

02:12

f of X 0 divided by X minus X 0 it's

02:16

the slope of the secant line through the

02:19

points X 0 f of X 0 and X f of X next we

02:25

need to let X approach X 0 and by doing

02:29

that the slope of the secant line will

02:31

change as this point X moves toward the

02:33

X 0 the slope of the red line that's the

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secant line will change here's an

02:38

example X moved closer to X 0 and the

02:41

line changed quite a bit as X moves

02:43

closer and closer and closer to X 0 this

02:46

secant line becomes a tangent line in

02:49

the limit in the limit X and X 0 kind of

02:51

overlap and the slope of the secant line

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becomes the slope of the tangent line in

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other words if this limit exists it is

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the slope of the tangent line to the

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graph of f at X 0 the derivative is thus

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the slope of the tangent line to the

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graph at X 0 the slope can be negative

03:15

it can be zero it can be positive here

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you see an example for the slope and

03:19

that's the derivative is negative the

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tangent line has a negative slope it's

03:23

sloping downward in the middle example

03:26

you see an example where the tangent

03:28

line is horizontal it has a zero slope

03:31

and therefore the derivative is zero at

03:33

this point X 0 where that line is

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tangent to the graph in this example

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here the slope of the tangent line is

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positive and therefore the derivative is

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positive and it becomes more positive as

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you move along this crap because the

03:45

line would get steeper and steeper the

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derivative does not have to always exist

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however here's an example where the

03:52

graph has some kind of corner so this is

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my point X 0 f of X 0 and I were to

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approach this point X 0 with X values

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from the left in these secant lines all

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look a little bit like this and

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definitely have a positive slope even

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the limit the slope is positive however

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if I approach this point from the right

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my secant lines have a negative slope

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and approach of limit it's negative and

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slope and they can not find a one

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tangent line to the graph of f at this

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why next zero there is no tangent line

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and so there is no unique slope which

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means this function does not have a

04:30

derivative at X 0 so in this example f

04:34

prime of x 0 does not exist let's return

04:46

to complex numbers now the derivative is

04:49

defined in a completely analogous way to

04:52

the real value case we say a complex

04:54

valued function f of a complex value is

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complex differentiable and typically we

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don't really say complex differentiable

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we say it's differentiable if it's clear

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from the context that we mean complex

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differentiable so function is

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differentiable at a point Z 0 which must

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be in the domain of F obviously if the

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limit of the difference quotient exists

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so we look at F of Z minus F of C 0

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divided by Z minus is 0 and let C

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approach C 0 now notice though if you

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pick a point Z 0 in the complex plane

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there are lots of ways of approaching

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that point it's not just along the real

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axis like we had to look at in the real

05:33

valued case you can also come in the

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complex direction from the imaginary

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axis you could come at a 45 degree angle

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or you could even approach in spirals

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there's lots of different ways of having

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Z approach C 0 and so for this limit to

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exist is a little harder to check then

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in the real gates if this limit exists

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it's the note at F prime of C 0

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sometimes we also use the notation DF DZ

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as a derivative of that with respect to

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Z at C 0 or just G DZ of F of Z at Z

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equal to 0

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let's look at some examples suppose f of

06:20

Z is a constant function - no matter

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what do you plug in you get the same

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constant value out of it if you prefer

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you're going to just think about a

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specific constant like 10 right here

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pick a Z 0 and C and look at this

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difference quotient what is f of Z minus

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F of C 0 divided by Z minus C 0 F of Z

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is this constant C but f of C 0 is also

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equals to C because F is a constant

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fraction divided by Z minus C Z or the

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numerator evaluates to 0 the denominator

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is nonzero so the quotient is equal to 0

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and as Z approaches to 0 it doesn't

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matter how Z approaches C 0 this

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quotient is always 0 completely has a

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limit 0 therefore the derivative of F is

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equal to 0 for all points in the complex

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plane sometimes instead of writing the

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difference quotient as f of Z minus F of

07:13

C 0 divided by Z minus C 0 we write Z as

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Z 0 plus h where H is a complex number

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and the difference quotient then would

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become instead of writing Z here we're

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gonna write Z 0 plus h so FX is 0 plus h

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minus f of C 0 and Z minus C 0 then

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becomes if we write Z as Z 0 plus h and

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subtract from that Z zero what's left

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over this H so this is the exact same

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quotient just written in a different way

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in terms of the difference H rather than

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in terms of the sea so the difference

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quotient is FX is 0 plus h minus f of c

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0 divided by h or often we just write f

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of Z plus h minus f FC over H when we

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get tired of writing is easier all the

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time and we'll take the limit as H

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approaches 0 because when Z and cz rated

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closer and closer to each other

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means their difference Z minus C 0 which

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is H goes to 0 so here it's another

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example let's look at the function f of

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Z equals Z so the function that assigns

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to each point just the same point so

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what is f of C 0 plus h minus f of C's

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/ h well i see 0 plus h is just 0 plus h

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which you see right here FX 2 0 is

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simply as easier f just returns exactly

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the argument that can pass into it we

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divide by h and you see in the numerator

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the z zeros cancel out with each other

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and you're left with H over h h / HS 1

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no matter what the value of ages and as

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h approaches 0 it remains 1 no matter

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how h approaches 0 which means this

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limit exists and that means the

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derivative of f is equal to 1 for all Z

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in the complex plane so this function is

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super rentable everywhere with

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derivative 1

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let's next look at the function f of Z

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equals Z squared the difference quotient

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then becomes F of Z zero plus h and the

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function f just takes its input in this

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case e 0 plus h and squares it so that

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becomes C 0 plus h quantity squared

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minus f of C 0 which is C 0 squared

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divided by H this term is e 0 plus h

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quantity squared multiplies through to C

09:43

0 squared plus 2 times C 0 times h plus

09:48

h squared and then we subtract C 0

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squared from this expression so we have

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a Z 0 squared here we subtract 1 here

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which means the numerator simplifies to

09:59

2 Z 0 h plus h squared next we can

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cancel an each out of the numerator each

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out of the denominator so we are left

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with 2 Z 0 plus h and as H goes to 0

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this plus H term becomes less and less

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important so that in the limit we are

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left with 2 times C 0 so the limit

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exists and therefore f prime of Z is 2 Z

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for all Z in the complex plane let's

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look at another example F of Z equals Z

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to the ends the function takes the

10:33

argument and raises it to the nth power

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therefore the difference quotient

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becomes F of Z 0 plus h which is Z 0

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plus h to the nth power minus F of Z 0

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which is C 0 to the N divided by H now

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how do you multiply at Z 0 plus H to the

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N power we can do that using the

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binomial theorem but we're actually only

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interested in the first few terms the

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binomial theorem says that this is equal

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to the sum and choose K times Z 0 to the

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K times H to the N minus K this runs

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from K from 0 to n you don't really need

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to know this binomial theory all you

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need to know is that when you multiply

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it through you have different powers of

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and different powers of agent always add

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up to n and binomial coefficients tell

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me how many of those terms you have so

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for example you'll have a Z zero to the

11:39

nth power and no each term then you have

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is easier to the n minus first power and

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an H term then as easier to the n minus

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second power and an H squared terms that

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would be is easier to the n minus third

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and H cubed and so forth all the way to

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Z zero to the no power times H to the N

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and the number of times these terms

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occur even by the binomial coefficients

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to 0 to the M occurs once patients

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easier than n minus one occurs n times

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with n choose one this coefficient here

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is n choose two and so forth none of

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these really matter for the limit in the

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limit the only terms that are really

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important are these first two terms and

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you noticed we have again as easier to

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the N term that are going to subtract so

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that term goes away we're left with

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terms that all have an H or higher power

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of H in them the only one that has a

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power one of H is this first term n

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times H times easier to the n minus one

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so if we cancel out an H of the

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numerator and denominator this term has

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no H's left in it becomes n times C 0 to

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the N minus 1 and all the other terms

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just lose one power of their H but all

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have an H left in them you could even

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factor out an H and have all these

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remaining terms left in here so as H

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approaches 0 we have some number here

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that gets multiplied by H which becomes

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smaller and smaller and smaller and in

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the end becomes negligible in other

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words the limit of this expression is

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anytime 0 to the N minus 1 therefore the

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function Z to the N is differentiable in

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the complex sense with derivative and

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times Z to the N minus 1 for all Z in C

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now that we have taken a few derivatives

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it would be great to understand how we

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can combine functions whose derivatives

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we already know so what happens she be

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differentiable functions and

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differentiable

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z and suppose H is another poach that's

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their eventual efficacy we want to look

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at the composition that's why we want to

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have age differential what efficacy and

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suppose C is a complex concept then the

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derivative of a constant times F is the

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same thing at the constant times the

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derivative of F so constants can go out

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to the side when taking derivatives next

14:05

the derivative of the sum of two

14:07

functions is the sum of the derivatives

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in other words you can individually

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differentiate these two functions and

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then just add the results together the

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product is a little bit more complicated

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it's not sufficient to just simply

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differentiate each function but there's

14:25

a product rule you differentiate F and

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multiply with G next you don't

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differentiate that and multiply that by

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the derivative of G this also rule for

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the derivative of the quotient F over G

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obviously you can't do that if G of Z is

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zero so you want to require here that G

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of Z is non zero but if it is nonzero

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the quotient rule is the same as it is

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in calculus you find the derivative of

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the quotient by squaring the denominator

14:55

writing down the denominator again times

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the derivative of the numerator - this

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time the numerator times the derivative

15:03

of the denominator so you almost see the

15:06

product rule in the numerator except for

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that minus sign and so you need to

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remember which term comes first which

15:13

term is the one with the minus sign and

15:15

which term is the one without the minus

15:16

I the way I remember that is I squared

15:19

the denominator and write it down right

15:20

away again and then times the derivative

15:23

of the numerator therefore I know which

15:24

one is the firm that comes first

15:26

finally the derivative of the

15:29

composition H of F of Z can be found by

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taking the derivative H prime of Z and

15:39

multiplying that by F prime of Z rather

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than showing you all these proofs which

15:45

are the same

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the proofs in calculus I'm going to

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show you a number of examples these are

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important rules to practice if you're

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not too familiar with these rules from

15:54

calculus anymore I highly recommend that

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you do some practice you could for

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example practice with all the problems

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I'm about to present but by trying to

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find the derivatives by yourself first

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before looking at my solutions

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let's start with a function f of Z

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equals five Z cubed plus two Z squared

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minus Z plus seven so we have a sum of a

16:23

bunch of functions a difference is the

16:25

same as the sound we can therefore

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differentiate these four functions

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individually so what are all these

16:32

derivatives the derivative of five times

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the cubed is a constant times e cubed

16:39

constants can go to the side five times

16:42

the derivative of Z cubed we just found

16:45

the derivative of z to the n that was n

16:48

times e to the N minus 1 in other words

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the derivative of Z cubed is three times

16:53

Z squared so the derivative of 5 C cubed

16:56

is 5 times 3 d squared similarly we find

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the derivative of 2z squared the 2 goes

17:03

to the side times the derivative of z

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squared which is 2z the derivative of z

17:10

which is either the one that we found

17:13

earlier it's was one horn could also

17:15

apply this asked the derivative is e to

17:17

the first power which is 1 times e to

17:20

the 0 power which is also 1 and second

17:23

is simply a constant whose derivative is

17:25

0 the airport I didn't write down 2 plus

17:28

0 part now if you wanted to simplify

17:30

this a bit because it's 15 Z squared

17:32

plus 4 z minus 1 next let's look at 1

17:37

over Z we can use the quotient rule for

17:40

that one even though there's also easier

17:43

ways of finding this to Rivet equipment

17:44

is find it using the quotient rule so we

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square the denominator is Eastway we

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write down the denominator again that Z

17:51

times the derivative of the numerator

17:53

which is 0 the numerator is a constant

17:54

its derivative is here

17:56

mine is the numerator 1 times the

17:59

derivative of the denominator the

18:01

derivative of Z is 1 the numerator

18:03

therefore simplifies to 1 with a

18:06

negative sign in front of it the

18:07

denominator is Z squared so the

18:09

derivative of 1 over Z is minus 1 over Z

18:12

squared a different way of finding this

18:16

derivative would have been to look at

18:18

this function as the

18:20

and of Z equals Z to the power negative

18:23

one and realizing that this rule for the

18:28

derivative of Z to the N also holds for

18:30

negative exponents and so the derivative

18:32

is therefore minus one times Z to the N

18:38

minus 1 so minus 1 minus 1 which is

18:41

minus Z to the minus 2 and if you

18:46

rewrite that that is minus 1/2 by the PI

18:49

Z squared let's look at F of Z equals Z

18:53

squared minus 1 to the N power so here

18:57

we have an inside function and another

19:00

function on the outside to the power n

19:02

we can use the chain rule the rule that

19:06

said that the derivative of H of F of Z

19:17

is H prime of F of C times F prime of C

19:23

so we need to find the derivative of the

19:25

outside function which is the function

19:27

that raises its argument to the nth

19:29

power or the derivative of that is n

19:31

times the argument to the n minus first

19:34

power and then multiply that by the

19:36

derivative of the inside is e squared

19:38

minus 1 the derivative of C squared

19:40

minus 1 is 2z via for the derivative in

19:43

this case is n times z squared minus 1

19:46

to the n minus first power times choose

19:49

let's look at another easy F of Z is Z

19:52

squared minus 1 times 3 is 3 plus 4

19:55

which is the product of two functions of

19:58

course we could multiply through and

20:00

then differentiate term by term but I

20:03

want to show you a use of the product

20:05

rule here the product will set that we

20:07

need to take the derivative of the first

20:09

function which in this case is 2z the

20:12

derivative of z squared minus 1 and

20:13

multiplied with the second function 3z

20:16

plus 4 plus the first function Z squared

20:19

minus 1 times the derivative of the

20:22

second function but the derivative of 3

20:24

Z plus 4 is simply 3 because 4 is a

20:28

constant function whose derivative is 0

20:32

finally let's look at Z divided by Z

20:36

squared plus 1 again let's use the

20:37

quotient rule which means we have to

20:39

square the denominator is e squared plus

20:41

1 quantity squared

20:43

write down that denominator again and

20:45

multiply it with a derivative of the

20:47

numerator but the numerator is Z its

20:49

derivative is 1 so I didn't even write

20:51

down the times 1 right here subtract

20:55

from that the numerator Z times the

20:58

derivative of the denominator which is 2

21:00

Z and if you wanted to simplify that we

21:03

have a Z squared plus 1 minus 2z squared

21:06

so the numerator becomes 1 minus Z

21:08

squared and the denominator is the same

21:11

as it was before ivory would it as 1

21:13

plus Z squared quantity squared

21:15

to make it look nice

21:22

here's a non-example a function that is

21:25

not complex differentiable f of Z equals

21:29

the real part of Z in other words if you

21:32

write Z as X plus iy them and of X plus

21:35

iy is simply the real part of what we

21:38

plug in so it's just the X let's use the

21:41

f of Z plus h minus f of Z over each

21:44

form of the difference quotient and

21:46

that's right HSH X the real part of H

21:48

plus IH y the imaginary part of H then

21:53

what is f of Z plus h Z plus h would be

21:58

X plus HX plus I Y plus hy and if we

22:03

only look at the real part of that that

22:05

real part is X plus HX we subtract from

22:09

that F of Z F of Z is X we see that

22:13

there is an X here and we subtract X

22:17

here so that cancels out the numerator

22:19

becomes simply HX the denominators H HX

22:24

was the real part of H so the difference

22:26

quotient is Cynthia the real part of H

22:28

divided by H what happens as H goes to 0

22:32

as we noted earlier there are lots of

22:36

different ways of approaching a point H

22:38

could go to 0 along the x axis or it

22:42

could go to 0 along the imaginary axis

22:45

Spier or along all kinds of other ways

22:48

but let's check out for starters what

22:51

happens when H approaches 0 along the

22:54

real axis so H in other words is real

22:56

it's of the form H X plus I times 0 does

23:00

not have an imaginary part in other

23:02

words the real part of H which is HX is

23:05

the same as H itself because there is no

23:07

imaginary part in other words in this

23:10

quotient right here the numerator is

23:13

equal to the denominator which means the

23:16

whole fraction evaluates to 1 and as H

23:19

goes to 0 just along the real axis the

23:22

limit is 1 so the limit exists as H

23:25

approaches 0 along the real axis next

23:29

let's have a ship roach serie along the

23:31

imaginary axis then an age has no real

23:34

part

23:35

and only has an imaginary part two ages

23:37

of the forum zero plus I times eight why

23:40

in other words the real part of H is

23:43

zero so again in this fraction real part

23:46

of H over H the numerator is zero so the

23:50

whole fraction evaluates to zero and in

23:53

the limit as H approaches zero along

23:55

that imaginary axis the limit is still

23:57

zero that's a different limit that we

23:59

obtained coming along the real axis

24:02

let's next look at a sequence H and that

24:05

approaches zero in a spiral type way HN

24:10

is I to the N over N we looked at the

24:13

sequence before we know the sequence

24:14

approaches zero what is the real part of

24:17

HN divided by H n then dividing by n

24:20

hakonsen both numerator and denominator

24:22

it's the airport cancel except the

24:24

numerator is real part of I to the N

24:27

over N the denominator is I to the N

24:29

over N which simplifies to real part of

24:31

I give it and override to then the I to

24:34

the N is 1 or negative 1 or I or

24:39

negative 5 and you look at the quotient

24:41

for even ends by the that end is real

24:44

and so the quotient is 1 for odd ends I

24:47

to the N is purely imaginary Ziya or

24:50

negative I and the quotient is therefore

24:53

0 because the real part of either than

24:55

and the syrup the sequences therefrom

24:57

bounces back and forth between 1 and 0

24:59

does not have a limit as n goes to

25:01

infinity because it does not approach

25:03

one of these numbers it does this baffle

25:05

work all together we see that this

25:08

function f is not differentiable

25:10

anywhere because the limit of the

25:11

difference quotient does not exist it is

25:14

different depending on how he approached

25:16

the origin some times the limit doesn't

25:18

even exist at all

25:19

so this function does not have a

25:22

derivative anywhere in the complex plane

25:30

here's another non-example an athlete

25:34

whose e equals the complex conjugate of

25:38

Z so what is f of Z plus h minus f of Z

25:41

over H when Z plus h get snap under F 2

25:48

Z plus h conjugate and we learned that

25:52

we can pull that inside that the

25:54

conjugate of Z plus the conjugate of H

25:56

and that's what you see right here we

25:58

subtract from that F of Z which is the

26:00

conjugate of Z and again we see there's

26:03

the conjugate of Z we subtract the

26:04

conjugate of Z those two cancelled out

26:06

and the numerator is just the conjugate

26:08

of H the denominator is H just I have a

26:12

limit as H goes to zero I don't think so

26:15

again if H is a real value than the

26:18

conjugate of H of simply H and the

26:21

quotient is equal to 1 approaches 1 as H

26:23

goes to 0 if H is purely imaginary then

26:27

the conjugate of H is the opposite of H

26:30

and so the quotient evaluates to

26:31

negative 1 and has a limit of negative 1

26:34

and if H approaches the origin any other

26:36

way it gets even more complicated

26:39

certainly the limits are not the same

26:41

and that already proves to us that this

26:44

limit of the difference quotient does

26:46

not exist as H approaches zero and

26:49

therefore f is not differentiable

26:51

anywhere in the complex plane here's a

26:54

fact it's the same as is true in real

26:57

value calculus if f is differentiable at

27:01

C 0 then it must be continuous at C 0

27:04

not the other way around a function can

27:06

be continuous without being

27:08

differentiable what if it happens to be

27:10

differentiable that's a stronger concept

27:12

than being continuous it implies

27:14

continuity here's how you would prove

27:16

suspect instead of showing that the

27:19

limit of f of z is equal to f of z 0

27:24

which is what we need to show for the

27:27

function to be continuous at C 0 well

27:31

bring this F of C 0 to the other side

27:33

and we look at F of Z minus F of C 0

27:35

which then needs to be equal to 0

27:37

so we look at C minus F of C 0 as Z

27:41

approaches C 0 and here comes the trick

27:44

we divide by Z minus C 0 well you can't

27:48

just do that but if we also multiply by

27:50

Z minus C 0 then we really just

27:52

multiplied f of C minus F of C 0 by 1 so

27:55

that's allowable but interpreting this

27:59

as the difference quotient times C minus

28:01

C 0 if the function f has a derivative

28:04

then the difference quotient has a limit

28:06

as C approaches to 0 that limit is F

28:08

prime of C 0 Z minus C 0 also has a

28:12

limit as Z approaches C 0 namely

28:14

0 so the limit is f prime of C 0 times 0

28:17

and have is 0 since f of Z minus F of C

28:22

0 approaches 0 as Z goes to Z 0 the

28:26

function f is therefore continuous at C

28:28

0 again a function Kennedy continues

28:32

without being differentiable but it

28:35

can't be differentiable without being

28:37

continues we'll see an example about

28:39

this shortly here's another definition a

28:43

function f is analytic in an open set in

28:46

the complex plane if it is complex

28:49

differentiable at each point in you we

28:54

need to prevent ability in a whole open

28:56

set we're not interested in functions

28:58

that are simply differentiable just at a

28:59

point we want them to be differentiable

29:01

in a whole open set if that opens head

29:04

happens to be all of the complex plane

29:07

so if the function is differentiable and

29:09

each point in the complex plane we've

29:12

seen some examples then we also say this

29:14

is an entire function here are some

29:17

examples polynomials are analytic and

29:20

see because we showed that anything of

29:23

the form Z to the N is differentiable

29:26

everywhere in the complex plane we can

29:28

even put a constant in front of it

29:30

that's still differentiable and then we

29:31

can act to that another term D times e

29:36

to the animal the different power and

29:38

more and more of these so these foreign

29:40

polynomials and therefore polynomials

29:42

are in

29:43

in the complex plane and therefore

29:45

entire functions were actual functions

29:48

which are polynomial divided by another

29:50

polynomial R by the quotient rule

29:53

analytic wherever we're not dividing by

29:55

zero so whenever the denominator is non

29:57

zero a rational function is also

30:00

analytic we've seen that the function f

30:03

of Z equals the complex conjugate of Z

30:06

is not differentiable anymore and so

30:09

it's also not analytic anywhere

30:11

similarly the real part of Z wasn't

30:14

differentiable anywhere so it's also not

30:16

analytic anywhere let's look at another

30:18

example FC is the absolute value of Z

30:22

squared remember how to find the

30:24

absolute value squared of a complex

30:26

number we had various ways that was

30:29

defined one way was to take x squared

30:32

plus y squared if Z was off the form X

30:37

plus iy but once we learned about the

30:41

complex conjugant we also learned that

30:43

this is equal to Z times the conjugate

30:47

of Z because Z times the conjugate of C

30:50

was X plus iy times X minus iy if you

30:56

multiply through this becomes x squared

30:59

minus I squared Y squared I squared is 1

31:03

that's 1 so we're back to x squared plus

31:05

y squared let's apply back in our case

31:08

here we need to look at the difference

31:09

quotient which is f of z plus h minus f

31:12

of Z divided by H F of Z plus h is the

31:17

absolute value of Z plus h squared F of

31:20

Z is the absolute value of Z squared and

31:22

now be use of that that the absolute

31:25

value squared can be obtained by taking

31:28

the complex number and multiplying it

31:30

with its conjugate the conjugate of Z

31:32

plus h is the conjugate of Z plus the

31:35

conjugate of H and you subtract from

31:37

that still Z squared when we multiply

31:40

through in this B plus h times Z

31:42

conjugate plus h cons you get we get as

31:45

Z times Z conjugate which is just the

31:48

absolute value of Z squared plus Z times

31:51

H conjugate which is here right here

31:54

plus h times

31:56

you conjugate this term plus h times H

31:59

conjugate and there's still the minus C

32:01

squared here again you see this is these

32:04

squared terms this is at it and then

32:06

subtract it so they cancel out

32:08

we're left with are these three terms in

32:11

the numerator divided by H the second

32:14

and the third term have an H in them so

32:16

you can cancel out that H the first term

32:20

however does not have an H and so we

32:23

cannot cancel anything out there what

32:26

we're left with therefore is a z bar

32:29

from this second term with the H cancel

32:31

that and an H bar from the third term

32:34

with each cancel but and Z times H bar

32:38

over H with H didn't cancel and naught

32:42

is the Troublesome term as H goes to 0 Z

32:48

bar is not affected that's just mean of

32:50

grievances bar H bar just goes to 0 as H

32:56

goes to 0 but H bar over H s we saw

32:59

earlier does not have a limit so unless

33:01

this term isn't there miraculously it

33:04

will cause trouble and will make the

33:06

limit not exist except when this Z is

33:10

equal to 0 therefore making this term

33:13

non-existent so if Z is non zero then

33:17

this limit of the difference quotient

33:19

simply does not exist because of the H

33:21

bar over H issue Z is equal to 0 that

33:25

issue is gone and the limit is simply 0

33:28

because Z is 0 z bar is also 0 so at 0

33:32

the limit of the difference quotient

33:34

exists so f is differentiable at 0 with

33:38

the derivative 0 it is not

33:41

differentiable anywhere else one point

33:44

is not enough to call the function

33:46

analytic we need the function to be

33:47

differentiable in the whole open set in

33:49

order for it to be called analytic so f

33:51

is not analytic anyway note however that

33:55

this function is a perfectly continuous

33:58

function it's continues everywhere in C

34:00

it's only differentiable at the origin

34:02

it's not analytic anywhere next up are

34:06

the Koshi Riemann equations

34:08

those are equations that relate the

34:11

derivatives up to U and V we write FS u

34:14

plus IV it relates the derivatives in

34:17

the X and the y direction of U and V

34:19

with each other I explained that in the

34:22

next lecture