Complex Analysis Midterm

Review Sheet

The midterm will be from 7-9 pm on Monday, April 24. It will consist of

10 definitions to state

5 Theorems to state

5 Theorems to (state and) prove

10 True/False questions to test facts we have learned that don’t have names and to test ability to check examples

Definitions: If *’d, be able to explain with a diagram.

1) Complex Numbers

2) Modulus (norm) of a complex number *

3) Real and imaginary parts of a complex number/function

4) Argument of a complex number, principle argument *

5) Polar coordinates *

6) Complex conjugate *

7) Open set in C *

8) Closed set in C

9) Limit point of a set in C *

10) Isolated point of a set in C *

11) Boundary of a set in C *

12) Limit of a function f:S à C at z₀ in S, where S is a subset of C

13) Continuity of a function f:S à C at z₀ in S, where S is a subset of C

14) Open subset of S, where S is contained in C, or, relatively open set in S.
15) Sum of paths in C

16) Opposite path to a path in C.

17) Path connected subset of C

18) Step path

19) Step path connected subset of C

20) Path components of a subset of C

21) Domain in C

22) Convergent and divergent series in C

23) Absolutely convergent series

24) Power series

25) Differentiable function f : Sà C

26) exp(z), sin(z), cos(z), sinh(z), cosh(z)

27) Integral of a complex function along a path in C

Theorems: If *’d, be able to explain with a diagram. If **’d, be able to prove.

1) The Complex numbers form a field

2) Triangle inequality, norm of product is product of norms **

3) Properties of complex conjugate

4) The complex numbers cannot be ordered **

5) A set S in C is closed if and only if it contains all of its limit points **

6) A function f:S à C, where S is a subset of C, is continuous if and only if
for all open sets U in C, f ^-1 (U) is open in S. **

7) Paving lemma **

8) An open path connected subset of C is step path connected **
9) Prop 2.11

10) A convergent sequence in C is bounded. **

11) Cauchy Criterion for convergence, or, General Principle of Convergence

12) An absolutely convergent series is convergent **

13) Comparison Test

14) Ratio Test

15) Radius of Convergence Theorem **

16) Product of absolutely convergent series

17) Cauchy Riemann Equations **

18) Partial converse to Cauchy Riemann Equations **

19) If f’=0 on a domain then f is constant there. **

20) Prop. 4.8 **

21) A power series may be differentiated term by term inside its

radius of convergence **

22) Complex exponential and trig functions are periodic **

23) The Fundamental Theorem of Contour Integration **

24) Antiderivative of a power series **

25) Estimation Lemma **

26) Theorem 6.11 **