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 donÕt have names and to test ability to check examples
Definitions: If *Õd, 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 ˆ C at z0 in S, where S is a subset of C
13) Continuity of a function f:S ˆ 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 : Sˆ 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 *Õd, be able to explain with a diagram. If **Õd, 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 ˆ
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 fÕ=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 **