5.1 • THE COMPLEX EXPONENTIAL FUNCTION 161(c) Precisely where should the images of the points ±i7r be located?- Verify Equations (5-3) and (5-4).
- Express e• in the form u +iv for the following values of z.
(a) -3•.
(b)
(c) - 4 + 5i.
(d)-l+i^3 ;.
(e) 1 + i^5 ;.
(f) i -2i.- Find all values of z for which t he following equations hold.
(a) e• = - 4.
(b) e• = 2 + 2 i.
(c) e' = J3 -i.(d) e• = - 1 +iV3.
6. P rove that I exp (z^2 ) I :5 exp (lzl^2 ) for all z. Where does equality hold?
- Show that exp (z + i7r) =exp (z - i7r) holds for all z.
- Express exp (z^2 ) and exp(~) in the Cartesian form u(x, y) +iv (x, y).
- Explain why
(a) exp (z) =exp z holds for all z.
(b) exp (z) is nowhere a nalytic.- Show that le- • I < 1 iff Re (z) > O.
- Verify that
(a) z -lim 0 •'-z^1 = 1.
(b) :;:-i1f lim e'±l Z-l ?f = -1.12. Show that f (z) = ze' is a nalytic for all z by showing that its real and imaginary
parts satisfy the Cauchy-Riemann sufficient conditions for d ifferentiability.- Find the derivatives of the following.
(a) e".
(b) z^4 exp (z^3 ).
(c) e<<>+•bl•.(d)exp(~).