Advanced Engineering Math II phần 3 pdf

21

Think of it as the above map in reverse: The above picture on the right shows the top half

the domain, and we get:

(F) f: CI!!ÆCI!!; f(z) = sin z

For this it is useful to remember that

f(x + iy) = sinx coshy + i cosx sinhy

and we find out that it does this

–π/2 π/2

Æ

–1 1

the next block over (π/2 ≤"x ≤"π) goes underneath the axis, and then it repeats as we go

across the left-hand

(G) f: CI!!-{0}ÆCI!!; f(z) = 1

z

or w =

1

z .

Look at what happens to the general point z = x + iy

w =

1

x"+"iy =

x"-"iy

x

2

"+"y

2 = u + iv

A vertical line in the w-plane corresponds to u = k

x

x

2

"+"y

2 = k, a constant

But this is the equation to a circle For instance, taking k =

1

2 gives the circle center (1, 0)

radius 1. In general, all these circles pass through the origin (where f is not defined).,

since the above equation, when cross-multiplied, is satisfied by (0, 0).

Similarly, horizontal lines also correspond to circles, but this time centered on the y-axis.

In general, we have the following: