Ejercicios Caughy Riemman

3.2 Cauchy-Riemann Equations

157

EXERCISES 3.2

Answers to selected odd-numbered problems begin on page ANS-13.

In Problems 1 and 2, the given function is analytic for all z.Show that the CauchyRiemann equations are satisfied at every point. 1. f (z) = z 3 2. f (z) = 3z 2 + 5z − 6i

In Problems 3–8, show that the given function is not analytic at any point. 3. f(z) = Re(z) 5. f (z) = 4z − 6¯ + 3 z 7. f (z) = x2 + y 2 4. f (z) = y + ix 6. f (z) = z 2 ¯ 8. f (z) = x y +i 2 x2 + y 2 x + y2

In Problems 9–16, use Theorem 3.5 to show that the givenfunction is analytic in an appropriate domain. 9. f (z) = e−x cos y − ie−x sin y 10. f (z) = x + sin x cosh y + i(y + cos x sinh y) 11. f (z) = ex
2

−y 2

cos 2xy + iex
2

2

−y2

sin 2xy

12. f (z) = 4x + 5x − 4y + 9 + i(8xy + 5y − 1)
2

13. f (z) = 14. f (z) = 15. f (z) =

y x−1 −i (x − 1)2 + y 2 (x − 1)2 + y 2 x3 + xy 2 + x x2 y + y 3 − y +i 2 + y2 xx2 + y 2 cos θ sin θ −i r r

16. f (z) = 5r cos θ + r4 cos 4θ + i(5r sin θ + r4 sin 4θ) In Problems 17 and 18, find real constants a, b, c, and d so that the given function is analytic. 17.f (z) = 3x − y + 5 + i(ax + by − 3) 18. f (z) = x2 + axy + by 2 + i(cx2 + dxy + y 2 ) In Problems 19–22, show that the given function is not analytic at any point but is differentiablealong the indicated curve(s). 19. f (z) = x2 + y 2 + 2ixy; x-axis 20. f (z) = 3x2 y 2 − 6ix2 y 2 ; coordinate axes 21. f (z) = x3 + 3xy 2 − x + i(y 3 + 3x2 y − y); coordinate axes 22. f (z) =x2 − x + y + i(y 2 − 5y − x); y = x + 2 23. Use (9) to find the derivative of the function in Problem 9. 24. Use (9) to find the derivative of the function in Problem 11. 25. In Section 2.1 wedefined the complex exponential function f (z) = ez in the following manner ez = ex cos y + iex sin y. (a) Show that f (z) = ez is an entire function. (b) Show that f (z) = f (z)