1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.1 • FUNCTIONS AND LINEAR. MAPPINGS 61

y

"'=f(l)

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Figure 2.11 The mapping w = f (z) = (-1 + i) z- 2 + 3i.

-------~EXERCISES FOR SECTION 2.1

1. Find f (1 + i} for the following functions.
   (a) f(z) =z+z-^2 +5.
   (b) I (z) = ,2~ 1.
   (c) f (z) = f(x+i y) =x+y+i(x^3 y - y^2 ).
   (d) f(z)=z^2 +4zz-5Re(z)+Im(z}.

v

2. Let f (z} = z^21 - 5z^7 + 9z^4 • Use polar coordinates to find

(a) /(- 1 +i).

(b) I (1 + i\1'3).

3. Express the following functions in the form u(x, y} +iv(x, y).

(a) f (z) = z^3 •

(b) f(z) = z^2 +(2- 3i)z.

(c) f (z) =fr.

4. Express the following functions in the polar coordinate form u (r, 8) +iv (r, 8).

(a) f(z) = zs +:zs.

(b) f(z) = z5 +:zs.

(c} Fbr what values of z are the above expressions valid? Why?

5. Let f (z) = f (x + iy} = e"' cosy+ ie"' sin y. Find
   (a) /(0).
   (b) I (i,,.}.