Z represents our previous oturmu

fyx of functions of two variables, such as

What future geometric meaning And their

that what they mean, and also the the simple linear

beyond the most simple and functional square z and y

but the intermediate compound of squares linear

function that squares with the xy as

basic de-frame sequence of z We saw functions.

Meanwhile, a special type of function we'll talk about.

They are composed of complex variables In short, let us remember complex functions:

minus 1 squared, square root, natural or It is not a real number.

These virtual numbers as a unit with IR Showing.

I still work these things before in western languages

I head for the arrival of imaginary words is derived from the letter.

Both x and y are real numbers here shown

I also created using the number of years as a Complex number is called.

This is the real part of the complex number x This

general shape representation, international representation of shape böyle.b

in the sense that the real meaning Imager

representation of the shape of the imaginary part of Z y is going on.

The circular coordinates in the space of a

If we take the point x and ys it with a vector

As such a vector in the plane We can show.

In this Cartesian coordinates xy As we will show theta

We can show the circular coordinates The plane is called demonstration et al.

Argan offering it for the first time anyone's name.

Of course, a very simple xy with rtet relationship.

Rn the R times the projection on the X axis cosine theta axis on .Y

so that the opposed projections theta axis

Run times on the projection cosine theta.

If we combine these two length as x + iy as

R sine cosine theta See X to R

R y de tete de la sine of theta, but I Because multiplying

Run the cosine common factor in brackets I come theta sine theta.

So the (Euler) Remembering the formula cosine theta plus sine theta E via IR

I like the circular içingör is dete coordinates

In this way a complex number can be shown.

They are very commonly used things.

All dynamic systems need solid of solids in liquids

dynamic events, in the electric field always in dynamic events

This is often used in complex numbers she has

from the outset to offer it for There are benefits.

Even in physics and atomic physics

size (Schrödinger) is Şurding in equation events

Again in this case defined by the IR complex representations come up.

When you think of a function only x plus a function consisting of IY.

As you can see this is a special function In a sense, the two değikenl

but on the other hand x IO free they do not move.

If x squared plus HT is now playing x is not free

Therefore when you get in free a univariate

function, but also the complex numbered That is a function of Zn Z

That is a function of the variable Z x plus We can show as iy.

This is called a complex-valued function.

In fact, we were the only one variable in the x and y We're going.

Therefore, each complex every change functions x and

fz as a general function of y can not be written.

more detail later on in this We'll see.

We will see here with examples.

Let's consider it as an example (w)

Let us define z as y have a double frame.

W, the value of these complex functions plus

Iv see here zn defined this as the frame of zn

plus the square of the first frame of the second The second frame

I squared y squared so as to compensate for income.

You can see the product of the two twice involved here, such as complex numbers.

as a dependent variable it real ie part of x squared

As you can see the square in the imaginary part x xy twice.

We have seen these functions.

XV yn which is the function of u

This kind of lines equal value consisted of hyperbolas.

If you look at the previous lesson right You will recall.

If V is here to give constant values you can see

Here again is the hyperbolic fit GB free hyperbolas occurs.

They are not leaving this kind of hyperbolic If you give a fixed value

for this kind of plus (parabola) the complex hyperbolic means

thus for negative values hyperbolas We have seen that hyperbolas.

Complex functions of two variables functions can be dealt with.

Have a special structure.

But when given as a function of Zn always two identical

functions of two variable functions We can reduce.

Show the circular coordinates It may be possible.

For example, although there logarithm z, zn circular

Over here in the coordinate expression r e A transport stream

As you can see it when positioned logarithm R

Once the conductor of E over a and b of this as the product of multiplying the

The logarithm of the sum of logarithms, but e to the logarithm of the conductor

Accordingly, the first sum of plus the logarithm of the logarithm of the second

Assuming no subsequent communication via e sure to Ryan.

Logarithmic and exponential functions of each other is opposite to

I just transport stream over to the logarithm theta remains.

As you can see this again plus Ivan structure means that u came to

For fixed values of the logarithms R

If r is equal to a constant fixed them returned.

r is a constant becomes hoops.

Here I was thinking of writing values because

As it is important to understand the nature.

Similarly, as you can see there is also the last was highly simplified.

Teta was.

Look theta hard to say.

When we choose b, constant theta In constant that x

axis making an angle theta with the constant geometric bodies.

These truths.

So this log, logarithm function z is confirmed.

Where two functions equivalent here

equal value, as indicated by the observed line if the lines.

Each complex function

as a function of z is an expression can not be told.

I really like the situation.

If the two variables x and y, that bi You can not convert variable.

We need to convert the two variables.

This is the second variable z again and z with a line called conjugate

i ia shown by replacing the negative the functions achieved.

Summing these two lines z plus our two x Half of the line to be x z plus z.

See also here and remove the interests of b i After I remove the x's are also going.

I will have two years left.

When we divide the two i y the following two i would be.

Wherein z is in the line difference z'yl The following ii hit i'yl

If you anchor i'yl above minus i times i that comes up as the minus sign.

So a return here, Denis, variable transformation of each case then

a function of x and y and z and z line can be expressed in terms.

In the simplest example, we can get function x

Although, x can not be expressed only in terms of z.

just like i y z expressed as can not be.

For example, we say x minus y i see you again here as

E, but this time in terms of z z line can be expressed.

Such a random function we take bi x squared plus i y

As E, di, when we calculate w x

from x to y squared plus i

z z plus one-half of the line

z to y squared minus z line

Put the square of e, take the following statement, You see.

We have had both our lines here as well.

So, for example, where all kinds of purpose complex function

expressed in a single z AmAyAcAğIydI be.

This kind of work functions in many bi useless.

While those expressed in only z might work.

Let Bi another example.

ie the real part of x cube minus three x y get square.

ie the imaginary part v's three x squared minus, times y minus three years to get the cube.

I wonder if it plus i v is such

written only in terms of a z I understand they can not be written.

Take it as you can see by combining plus i have it once.

But when we see this, edit the e, i is the square of a negative.

Where y is the square.

So this second term, three times x i y squared.

I already have here.

When i put it here once i y is there.

i y x squared times three.

Here are the minus sign.

If we take the square of the cube minus sign from the i minus income.

BI gives one the rest i also i here.

When we organize this term You can easily see

a plus b to the cube of type.

A x B of the cube which is also good if we take y plus three times the cube of its first

times the square of the first to the second term, We see here.

Plus three times the square of the first

times and to the second, the second term of the cube is coming.

This gives the term.

So three of these functions z he can produce.

So not only is a function of z

capable of producing a function, unless possible it happens.

So in that f z as w is We can write.

We do not want to go further in this entry.

Because these complex functions of their an important issue alone.

It utilizes the bivariate will consider the scope of functions again.

As you can see a special structure two type of variable functions.

It now faces in the next section We assume that diagnosis.

Outlines the surfaces of two variables We assume that we understand the surface.

There are of course infinitely many surfaces.

Bun, but with the main surfaces and a pound created surfaces.

In what follows two general variable

derivative and integral We will see them three

and will generalize to the multivariate.

You'll see a bivariate Once you understand them

of course, by drawing, geometry even more We understand comprehensive.

But these three, and even N, N, 50, 100, 1000, five, eight desired

Number conceptually including We will also see that it is difficult to generalize.

Bye for now.

Do not forget to study your lessons.

Mathematics B at the end of a single loaded is not something that can be done, unfortunately.

Perhaps the power is already there.

Every time they step on, step by step

If you go to work and you'll see s easy math will be something extraordinary.