MAT 461 Applied Complex Analysis /100
November 12, 1999 Test 2 name
Justify all major steps that involve substantial complex
analysis reasoning. On the other hand, there is no need
for lots of detail in steps that involve only calculus or
1234
25 25 25 25
algebra – often a computer print-out may be adequate documentation. You may use MAPLE throughout,
but it is YOUR responsibility to demonstrate that you have mastered the new material of this class.
1. Let CRbe the circle with center 0 and radius R > 0 traversed counter clockwise.
a. Evaluate the integrals RCR¯z dz and RCR¯z−1dz.
b. Explain what is wrong with the following reasoning:
“Since RCR¯z−1dz = 0 for every circle CRas in part a., f(z) = ¯z−1is entire by Morera’s theorem.”
2. Let Cbe the circle with center 0 and radius 5 traversed clockwise, and let C+and C−be the
circles with centers ±i, respectively, both with radius 1 and and both traversed counterclockwise.
a. Let f(z) = (z2+ 1)−1sin z. Explain why RCf(z)dz =RC+f(z)dz +RC−
f(z)dz. Show details!
b. Evaluate RCf(z)dz. (You may use part a.)
c. Explain why f(z) has an antiderivative defined wherever f(z) is analytic, or explain why it does not.
3. Let z0∈Cand r > 0. Suppose that fis analytic inside and on the circle Cr={z∈C:|z−z0|=r}.
a. Use Cauchy’s integral formula to show that f0(z0) = (2πr)−1R2π
0e−iθ f(z0+reiθ )dθ
b. Suppose that f0(z0) = 5i.
Explain why on every circle Crthere must be a point won the circle Crsuch that |f(w)| ≥ 5r.
Bonus: What does this imply about f(z0)? (Trick question – very easy answer.)
4. (Take home part of test 2, due at the beginning of the class on Monday November 15.)
Summarize in (preferrably) no more than two pages what Cauchy’s theorem is all about.
Suggestions: Include a precise statement of the theorem, an outline of a possible proof (highlighting
where the hypotheses come into play), typical applications and consequences, and a personal view
whether (why?) the theorem is a big deal (or not) (e.g. compare with the real case).
Ground rules: Work alone – include a statement that you worked alone, and sign your work.
You may use the text, or other books, but must properly acknowledge any sources used.
