Complex Analysis

Question 1

Triangle inequalities

Answer 1

Question 2

Define an open disc

Answer 2

Question 3

Define an interior point of S

Answer 3

Question 4

Define an open subset

Answer 4

A subset S of C is open iff each of its points is an interior

Question 5

Define a closed subset

Answer 5

A subset S of C is closed iff its compliment C\S is open

Question 6

Define a compact subset

Answer 6

A subset S of C is compact iff it is closed and bounded

Question 7

State the Completeness proposition

Answer 7

Every Cauchy sequence in C converges

Question 8

Define a continuous function

At a point, and on S

Answer 8

Let S be a subset of C, and f : S -> C be a function.
f is continuous at z0 in S iff lim(z->z0) f(z) = f(z0), i.e.

f is continuous on S iff it is continuous at every z0 in S

Question 9

Define a differentiable function at a point

Answer 9

Let S be open
f is differentiable at z0 in S iff the limit exists

This limit is the derivative of f at z0, denoted f'(z0)

Question 10

Define a holomorphic function

Answer 10

f is holomorphic on S iff it is differentiable at every z0 in S

Question 11

Define an entire function

Answer 11

If f is holomorphic on S (differentiable at every z0 in S),
and S = C,
then f is an entire function

Question 12

State the Cauchy-Reimann equations, and its assumptions

Answer 12

Let S in C be open, z0 in C, f : S -> C.
Write f(z) = u(z) +iv(z), z = x + iy.
If f is differentiable at z0, then the four partial derivatives exist at z0 with:

If the partial derivatives exist and are continuous near z0, then f is differentiable at z0.

Question 13

Define the integral of a continuous function over an interval [a,b]

Answer 13

Let [a,b] be an interval in R, and G : [a,b] -> C be a continuous function.
For G(t) = x(t) + iy(t),

Question 14

Define a path
State what a closed path is

Answer 14

A path C in C is a continuous function g : [a,b] -> C, from an interval [a,b] in R to C.

The path is closed if g(a) = g(b)

Question 15

Define a contour
In this case, define the length of C

Answer 15

A path C in C, g : [a,b] -> C, is a contour iff there are finitely many points (a = t0) < t1 < ... < (tn = b) in [a,b],
s.t. x(t) = Re( g(t) ) and y(t) = Im( g(t) ) have continuous derivatives in each subinterval (tk,tk+1) for k = 0, …, n-1.

The length of C is:

Question 16

Define the integral of a continuous function over a contour C

Answer 16

Let C be a contour in C, a continuous function g : [a,b] -> C from some interval [a,b] in R to C.
Let f : C -> C be a continuous function, then:

Question 17

State the Estimation lemma

Answer 17

Let C be a contour, and f : C -> C a continuous function.
Suppose |f(z)| <= M for all z in C, then:

Question 18

State the ‘Fundamental Theorem of Calculus’ and associated Corollary

Answer 18

Let S be an open subset of C, and C be a contour in S from z0 to z1.
Let F : S -> C be a holomorphic function, and assume f := F’ is continuous. Then:

Corollary: If C is closed, then integral equals 0

Question 19

Define a convex subset

Answer 19

A subset S of C is convex iff, for any two points z0, z1 in S. the straight line [z0,z1] between z0, z1 belongs to S

Question 20

State the power series P(z)

Answer 20

Let a0, a1, ... in C, and z0 in C, then the power series P(z) is:

The usual convergence tests (ratio test, M-test) apply

Question 21

State the radius of convergence of a power series P(z), and its results

Answer 21

P(z) converges absolutely and uniformly on D,r'(z0) for any r' < r.
P(z) diverges for any z in C s.t. |z - z0| > r.

Question 22

Derivative of the limit function of a power series

Answer 22

The limit function P(z) is holomorphic on D,r(z0), and its derivative is:

Question 23

Define the exponential function, and state its three properties

Answer 23

Question 24

Give the definition of Arg(z)

Answer 24

Question 25

Define the Trigonometric functions

Answer 25

Question 26

Define the principal value of logarithm

Answer 26

Question 27

Define the principle value of powers

Answer 27

Question 28

Define the sin and cos addition formulae

Answer 28

Question 29

Derivative of `a^z`

Answer 29

`(a^z)' = Log(a) a^z`

Question 30

Conversion between sin/cos and sinh/cosh

Answer 30

`sinh(z) = -i sin(iz)` `cosh(z) = cos(iz)`

Question 31

State Goursat's Theorem

Answer 31

Question 32

State Cauchy's Theorem

Answer 32

Let S in **C** be open and convex, and f : S -> **C** be holomorphic. Let C be any closed contour in S, then:

Question 33

State Cauchy's Integral Formula

Answer 33

Question 34

Define a Generalised Geometric Series

Answer 34

Question 35

Define the Taylor series of a function

Answer 35

Let S in **C** be open, and f : S -> **C** be holomorphic. Then f is infinitely often differentiable, and for every `z0` in S, the Taylor series is:

Question 36

State Cauchy's Integral Formula for the nth derivative

Answer 36

Let S in **C** be open, and f : S -> **C** be holomorphic. For a closed disk D in S, s.t. `z0` in D\ `delta`.D:

Question 37

State Morera's Theorem

Answer 37

Question 38

State Cauchy's Inequalities

Answer 38

Let S in **C** be open, and f : S -> **C** be holomorphic. For a closed disk D in S, s.t. `z0` in D\ `delta`.D:

Question 39

State the Maximum Principle

Answer 39

If `|f|` has any local maximum, then f is constant

Question 40

State Liouville's Theorem

Answer 40

Every bounded entire function is constant

Question 41

State the Fundamental Theorem of Algebra

Answer 41

Every non-constant polynomial function (with complex coefficients) has a root in **C**

Question 42

Define a Laurent series

Answer 42

A Laurent series at `z0` in **C**, with `an` in **C** for all n in **Z**, is of the form:

Question 43

What does it mean for a Laurent series to converge

Answer 43

A Laurent series converges to `f(z)` iff both series converge to `g(z)` and `h(z)`, and `f(z) = g(z) + h(z)`

Question 44

Define an (isolated) singularity of f

Answer 44

Let f be a function defined and holomorphic on an open disk D, centred at `z0` in **C**, but not at `z0`. Then `z0` is an (isolated) singularity of f, and f can be represented by a Laurent series: The coefficient `a-1` is the residue of f at `z0`, denoted `res(f, z0)`

Question 45

State the values of `an` in a Laurent series for each singularity `z0`

Answer 45

Question 46

State Reimann's theorem on removable singularities

Answer 46

Let f be a function defined and holomorphic on an open disk D, centred at `z0` in **C**, but not at `z0`. Then `z0` is an (isolated) singularity of f, and:

Question 47

State the criteria of a pole `z0`

Answer 47

Let f be a function defined and holomorphic on an open disk D, centred at `z0` in **C**, but not at `z0`. Then `z0` is an (isolated) singularity of f, and:

Question 48

State the criteria of an essential singularity `z0`

Answer 48

Let f be a function defined and holomorphic on an open disk D, centred at `z0` in **C**, but not at `z0`. Then `z0` is an (isolated) singularity of f, and: `z0` is an essential singularity of `f` iff `f(z)` does not converge to `inf` or any finite value as `z -> z0`

Question 49

Defien the winding number of C

Answer 49

Let C be a closed contour in **C**. For any `a` in **C**\C, the winding number of C around `a` is: `n(C,a)` is an integer

Question 50

State the Residue Theorem

Answer 50

Let S in **C** be open and convex, and C be a closed contour in S. Let f : S\{`a1,...,am`} -> **C** be a holomorphic function, with isolated singularities `a1,...,am` in S not lying on C. Then:

Question 51

State the Corollary of the Residue Theorem for an integral from `-inf` to `inf`

Answer 51

Let `R(z)` be a rational function without any poles in **R**. And such that the degree of the denominator is bigger than the degree of the numerator by at least 2. Then:

Question 52

Define a Meromorphic function

Answer 52

Let S in **C** be open, then a meromorphic function on S is defined and holomorphic on S, except for a discrete set of poles `a0,a1,...` of f. In this case:

Question 53

Define the principal part at `a` of a meromorphic function

Answer 53

Let S in **C** be open, and f a meromorphic function on S. Given any `a` in **C**, and any polynomial function `p(z)` with zero constant term, the function `h,a(z) := p(1/[z-a])` is the principal part at `a`.

Question 54

Define a collection of principal parts on S of a meromorphic function

Answer 54

Let S in **C** be open, and f a meromorphic function on S. A collection of principal parts on S consists of a discrete set of pairwise distinct points `a0,a1,...` in S, with a principal part `h,v(z)` at every `a,v`.

Question 55

State what it means for a meromorphic function to solve a given collection of principal parts

Answer 55

A meromorphic function f on S solves a given collection of principal parts on S iff its collection of principal parts is equal to the given one.

Question 56

Define what it means for a series of meromorphic functions to converge compactly

Answer 56

Let S in **C** be open. A series `Sum(v=1)(inf) fv` of meromorphic functions `fv` on S converges compactly on S iff for every compact subset K of S, there is a `v0` s.t. none of the poles of `fv` for `v >= v0` belongs to K, and s.t. `Sum(v>=v0) fv` converges uniformly on K.

Question 57

State Mittag-Leffler's Theorem

Answer 57

Every collection of principal parts on **C** is solvable. More precisely: Let `a0,a1,...` be a discrete set of pairwise distinct points in **C** s.t. `(0=|a0|) < |a1| <= |a2| <= ...`, and for each `v` let `h,v(z)` be a principal parts at `a,v` (`h0 = 0` allowed). Then: * Choosing the function `Pv`, `v >= 1`, to be the Taylor polynomial of `h,v` at 0 of sufficiently high degree, the series converges compactly on **C**. * If the functions `Pv` are chosen to be any entire functions s.t. the series converges compactly on **C**, then `h` solves the given collection of principal parts

Question 58

State the Partial fraction expansion corollary

Answer 58

Question 59

State the requirement and solution to:

Answer 59

For `z` in **C**\ **Z**:

Question 60

State the Partial fraction expansion of Cotangent

Answer 60

For `z` in **C**\ **Z**:

Question 61

Define what it means for a function to have a zero of order `n` in **N** at `a`

Answer 61

Let `a` in **C**, and f be a function that is defined and holomorphic on an open disk containing `a`. Then f has a zero of order `n` in **N** at `a` iff its Taylor expansion at `a` is of the form: where `an != 0`

Question 62

Define a collection of zeroes

Answer 62

A collection of zeroes in **C** is a set of pairwise distinct points `a0,a1,...` in **C**, (satisfying `lim(v->inf) |a,v| = inf` if set is infinite) together with an integer `n,v` in **N+** for every `v` (the order)

Question 63

Define a divisor, `div(f)`

Answer 63

The divisor `div(f)` of an entire function f, is the collection of its zeroes and corresponding orders

Question 64

State what it means for an entire function f to solve a given collection of zeroes

Answer 64

An entire function f solves a given collection of zeroes in **C** iff `div(f)` is equal to the given collection

Question 65

What does it mean if the entire functions `f` and `g` solve the same collection of zeroes

Answer 65

If the entire functions `f` and `g` solve the same collection of zeroes, then there exists an entire function `h` s.t., for all `z` in **C**: `f(z) = g(z) . exp( h(z) )`

Question 66

What does it mean about `f'/f`, if the entire function `f` solves a collection of zeroes

Answer 66

If the entire function `f` solves the collection of zeroes `(a1,n1),(a2,n2),...`, then `f'/f` is a meromorphic function on **C** that solves the collection of principal parts:

Question 67

State the Weierstrass Product Theorem

Answer 67

Every collection of zeroes in **C** is solvable. More precisely: * Let `(a0,n0),(a1,n1),...` be a collection of zeroes in **C**, s.t. `(0=|a0|) < |a1| <= |a2| <= ...`, (`n0 = 0` allowed). * For each `v >= 1`, choose `k,v` s.t.: * Then the infinitie product is a well-defined and entire function that solves the given collection of zeroes:

Question 68

Define the infinite product, and how it converges

Answer 68

... In general, the infinite product converges iff at most finitely many `a,v` vanish and the product (without `a,v = 0`) converges. (If an `a,v` vanishes, its value is defined to be 0).

Question 69

State the exponential relationship between a sum and product of sequences in **C**

Answer 69

Question 70

State the exponential relationship between a sum and product of sequences of functions on a subset S of **C**

Question 71

State Euler's Product Expansion of Sine

Answer 71

For `z` in **C**:

Question 72

Principal argument, exponential form

Answer 72

Question 73

`|z|`

Answer 73

Question 74

Exponential value of `i`

Answer 74

Question 75

Composition rules for entire functions

Answer 75

* Polynomials, `z`, and `exp` are entire functions * Compositions, products, and summations of entire functions are entire

Question 76

Analytic function

Answer 76

A function is analytic on a region, if it is differentiable at every point in the region => satisfies the Cauchy-Riemann equations