00:01
Let's look at the question.
00:02
We have to determine the analisability of the complex function.
00:06
So we need to check if it's satisfied cauchy -riemann's equation.
00:10
The cauchy -riemann's equation are set of two partial differential equation that complex the function satisfy that our f set is equals to u x y plus i v x y where z is equals to x plus i y and u v are the real number real value functions of x and y.
00:29
So the cauchy -riemann equation follows the partial differentiation of u by x is partial differentiation of v by y that is our equation one.
00:42
It's our equation two.
00:44
Now let's apply this equation to given function for the part a we have been given as f z is equals to z cube.
00:53
Let's see here u x y is equals to x cube minus 3 x y and v x y is equals to 3 x y y minus y cube.
01:11
Now using the cauchy -riemann's equation partial differentiation with respect to x will be 3 x y minus 3 y square and partial differentiation of respect to y will be 3 x square minus 3 y square minus 6 x y and 6 x y.
01:35
So the equation is satisfied which means that our x z is equals to z cube is analytic.
01:46
Now for the b part we have been given as our f z is equals to 1 by z.
01:52
So we'll see our u x y is equals to x by x square plus y square and v x y is minus y by x square plus y square.
02:10
Using cauchy -riemann's equation we have partial differentiation of x is equals to y square minus x square by x square plus y square whole square x square minus y square by x square plus y square whole square and minus 2 x y by x square plus y square.
02:38
There partial differentiation of y is 2 x y by x square plus y square.
02:49
The equation are satisfied everywhere except the origin 0 0 which is denominator becomes 0.
02:56
Thus fz is equals to 1 by z is analytic everywhere except 0.
03:16
Now let's see for c part where we have fz is equals to re z is equals to x.
03:24
Here we have u x y as x and v x y as 0...