62 CHAPTER 2 • COMPLEX FUNCTIONS
(c) f (i^2 ;).
(d) /(2+i7r).
(e) f (37ri).
(f) Is f a one-to-one function? Why or why not?- For z #0, let f(z) = f(x+iy) = ~ ln(x^2 +y^2 ) +ia.rctan~. Find
(a) /(1).
(b) f ( v'3 + i).
(c) f (1 + i,/3).
(d) I (3 + 4i).
(e) Is f a one-to-one function? Why or why not?- For z # 0, let f (z) = lnr + ifJ, where r = lzl, and fJ = Argz. Find
(a) /(1).
(b) / (-2).
(c) f(l+i).(d) f(- v'3+i).
(e) Is fa one-to-one function? Why or why not?
- A line that carries a charge of ~ coulombs per unit length is perpendicular to the
z plane and passes through the point .zo. The electric field intensity E (z) at the
point z varies inversely as the dist ance from zo and is directed along the line from
zo to z. Show that E (z) = .,.~.,. 0 , where k is some constant. (In Section 11.11 we
show t hat, in fact, k = q so that actually E (z) = ~-)
9. Use the result of Exercise 8 to find the points z where the total charge E (z) = 0
given the following conditions.(a) T hree posit ively charged rods carry a charge of ~ coulombs per unit
length and pass through the points 0 , 1 -i, and 1 + i.
(b) A positively charged rod carrying a charge of ~ coulombs per unit
length passes through the point 0, and positively charged rods carrying
a charge of q coulombs per unit length pass through the points 2 + i
and - 2+i.- Suppose that f maps A into B, g maps B into A, and that Equations (2-3) hold.
(a) Show that f is one-to-one.
(b) Show that f maps A onto B.11. Suppose f is a one-to-one mapping from D onto T and that A is a subset of D.(a) Show that f is one-to-one from A onto B, where B = {! (z) : z E A}.