MCS 321 EXAM 2
- What: Exam on integration
- When: 12:30-1:20 April 2, 2004
- Where: Olin 319
This will be a closed-book exam focusing on chapter 4 of our text.
You may use one new 4"-by-6" note card with anything you want written
on both sides, and you may use your Exam 1 note card as well.
Many of the problems will be similar to problems you did for homework.
Most problems will be of the form "evaluate this integral."
Main topics
- Integration of a complex-valued function of a real variable
Fundamental Theorem of Calculus in this case: p. 114 (4)
- Contours and contour integrals: definitions and calculations
- An upper bound on the modulus of a contour integral
- Theorem: Let f be continuous on domain D;
f has an antiderivative in D
iff all integrals of f along contours in D are path independent
iff all integrals of f around closed contours in D are 0.
Fundamental Theorem of Calculus in this case: p. 136 (1)
- Cauchy-Goursat Theorem: If f is analytic on and inside a simple
closed contour C, then its integral around C is zero.
- Tricks with contours
- Reversing direction:
an integral along C is cancelled by an integral along -C.
- Principle of deformation of paths, p. 152
- Cauchy integral formula
- Theorem: If f is analytic, then f has derivatives of every order.
Formula: p. 161 (4).
Handy integration formula: p. 161 (5)
- Neat and curious consequences:
- The only bounded entire functions are the constant functions.
- The Fundamental Theorem of Algebra (proved by complex analysis!)
- The Maximum Modulus Principle: A nonconstant analytic
function f does not have a maximum modulus in a domain D, but
its modulus will attain a maximum on the boundary B
if f is continuous on R := D union B and R is closed and bounded.