296 CHAPTER 12. THEOREMS INVOLVING LINE INTEGRALS

imaginary parts of f , f (z)z′ = ux′−vy′+ i(vx′+uy′) . Show that the real part of thiscontour integral is

∫C udx−vdy and the imaginary part is

∫C vdx+udy. Apply Greens

theorem and Problem 7 to conclude that this contour integral is 0. This is the Cauchyintegral theorem which is from Cauchy in the early 1800’s and is the foundation forcomplex analysis. This is roughly the way Cauchy did it.

11. Suppose f is analytic, explain why f̄ will usually not be analytic. f̄ (z) = u(x,y)−iv(x,y) where f (z) = u(x,y)+ iv(x,y).

12. Let C be a piecewise smooth oriented curve in C and let f : C→ C be continuousand bounded so that | f (z)| ≤M for some M. Show that |

∫C f (z)dz| ≤ML where L

is the length of this curve. For C an oriented curve, let −C be the same set of pointsbut oriented in the opposite direction. Explain how

∫C f dz =−

∫−C f dz. Go right to

the definition, −C involves t going from b to a where the interval for the parameteris [a,b].

13. Consider the following picture which illustrates a region for Green’s theorem U andinside a small disk of radius r called Ur centered at a which also is a region forGreen’s theorem. The boundaries of these two are oriented as shown.

U

C

Ur

a Cr

Show that the small circle is parametrized by a+re−it for t ∈ [0,2π]. Then justify thefollowing for f (z) = u(x,y)+ iv(x,y) , analytic on an open set containing Ū . Firstof all show z→ f (z)

z−a is analytic on the region between U and Ur. Next verify that thisregion is one which works for Green’s theorem.∫

C

f (z)z−a

dz+∫

Cr

f (z)z−a

dz = 0

Then show that limr→0

∣∣∣∣∫Cr

f (z)−( f (a)+ f ′(a)(z−a))z−a dz

∣∣∣∣= 0 using the differentiability of

f and the estimate of Problem 12. Thus∫

Cf (z)z−a dz+

∫Cr

f (a)+ f ′(a)(z−a)z−a dz= e(r) where

limr→0 e(r) = 0. Now show that limr→0∫

Crf (a)+ f ′(a)(z−a)

z−a dz = −2πi f (a) . Explain

why f (a) = 12πi∫

Cf (z)z−a dz. This is the famous Cauchy integral formula.

14. Use whatever convergence theorem is useful to show that in the above situation,

f (n) (a) =n!

2πi

∫C

f (z)

(z−a)n+1 dz

Also show that if f (z) is analytic on all of C (entire) and f ′ (z) = 0 for all z then f (z)is a constant. Show using the formula of this problem and the estimate of Problem12 that if f is bounded and entire, then f is a constant. This is Liouville’s theorem.

15. The easiest proof of the fundamental theorem of algebra which states that every non-constant polynomial having complex coefficients has a zero comes from the aboveLiouville’s theorem. If p(z) is a nonconstant polynomial with no zeros, explain why