Complex Variables Cathal Ormond
Contents 1 Introduction 3 1.1 Definition: Polar Form.............................. 3 1.2 Definition: Length................................ 3 1.3 Definitions..................................... 3 1.4 Lemma: The Minkowski Inequality....................... 4 1.5 Definition: Open Disk.............................. 4 1.6 Definition: Open Set............................... 4 1.7 Definition: Closed Set.............................. 4 1.8 Definition: Connected Set............................ 4 1.9 Definition: Convergence............................. 4 1.10 Definition: Cauchy Sequence.......................... 5 2 Limits and Continuity 6 2.1 Definition: Limits................................. 6 2.2 Definition: Limit Points............................. 6 2.3 Definition: Continuity.............................. 6 2.4 Definition: Uniform Convergence........................ 6 2.5 Theorem...................................... 7 2.6 Theorem...................................... 7 2.7 Definition: Partial Sum............................. 7 2.8 Definition: Power Series............................. 7 2.9 Definition: Radius Of Convergence....................... 8 3 Holomorphic Functions 9 3.1 Definition: Differentiability........................... 9 3.2 Definition: Holomorphic............................. 10 4 Integration 11 4.1 Definition: Path................................. 11 4.2 Definition: Path Integration........................... 11 4.3 Definition: Length................................ 11 4.4 Theorem...................................... 11 4.5 Theorem: Cauchy s Theorem.......................... 11 4.6 Theorem: Cauchy-Goursat s Theorem..................... 12 4.7 Definition: Residue................................ 12 4.8 Theorem: Cauchy s Integral Formula...................... 12 4.9 Theorem...................................... 13
Chapter 1 Introduction 1.1 Definition: Polar Form For all z = x + iy, we let x = r cos θ and y = r sin θ, then we say that the polar form of z is given by z = re iθ. Note r 0 and 0 θ 2π. 1.2 Definition: Length Given any z C with z = x + iy, we define the length of the number by z = x 2 + y 2. 1.2.1 Properties If we define d : C C R by d(z 1, z 2 ) = z 1 z 2, we see that (C, d) forms a metric space. 1.3 Definitions We make the following definitions for all z C, z = x + iy: e z = e x (cos(y) + i sin(y)) arg(z) = {θ + 2πk k Z} Arg(z) = {θ + 2πk 0 π < θ + 2πk 0 π} n re iθ = n re log(z) = ln z + i arg(x) Log(z) = ln z + iarg(x) cos(z) = eiz + e iz 2 sin(z) = eiz e iz 2i θ + 2πk! i n for 0 k < n
4 1.3.1 Properties From the above definitions of the exponential function, we can see that: e iθ 1 e iθ 2 = e i(θ 1+θ 2 ) eiθ 1 e iθ 2 = ei(θ 1 θ 2 ) 1.4 Lemma: The Minkowski Inequality If {r 1, r 2,..., r n } and {s 1, s 2,..., s n } are complex numbers, the: n (r j + s j ) 2 n (r j ) 2 n + (s j ) 2 j=1 1.5 Definition: Open Disk j=1 The open disk of radius r about some complex number a is given by r (a) = {z C z a < r}. 1.6 Definition: Open Set A set U C is called open if for all a U, there exists some r > 0 such that r (a) U. 1.7 Definition: Closed Set A set U C is called closed if C \ U is an open set. 1.8 Definition: Connected Set A set S C is called connected if for all U S with U both open and closed, U = or U = S. 1.9 Definition: Convergence We say that the complex sequence (z n ) converges to a number z if for all ɛ > 0 there exists some N N such that z n z < ɛ for all n > N. 1.9.1 Properties If (z n ) converges to z and (w n ) converges to w, then we have the follwing: (z n + w n ) converges to z + w (z n w n ) converges to zw If (w n ) and w are all non-zero, then ( zn w n j=1 ) converges to z w
5 1.9.2 Theorem If (z n ) is monotonically increasing and bounded above, or monotonically decreasing and bounded below, then it converges. 1.9.3 Theorem: Bolzano Weierstrass Every bounded sequence of complex numbers has a convergent subsequence. 1.10 Definition: Cauchy Sequence A sequence (z n ) is called cauchy if for all ɛ > 0, there exists some N N such that x n x m < ɛ for all n, m > N. 1.10.1 Theorem: Cauchy s Criterion in C A complex sequence is cauchy iff it is convergent.
Chapter 2 Limits and Continuity 2.1 Definition: Limits A function f : C C is said to tend to a value L as the point x approaches a if for all ɛ > 0, there exists some δ > 0 such that 0 < x a < δ implies that f(x) L < ɛ. 2.2 Definition: Limit Points A complex number is a limit point of a set A if for all ɛ > 0, there exists some z A such that 0 < z a < ɛ. 2.3 Definition: Continuity A function f : C A C is said to be continuous at a point z 0 if either z 0 is not a limit point of A or lim z z0 f(z) = f(z 0 ). If f is continuous at all points in A, we say that f is continuous. 2.3.1 Properties If f and g are continuous functions, then so are f + g, fg, f g, and if g is non-zero, f g. 2.4 Definition: Uniform Convergence A sequence of complex-values functions (f n ) defined on a complex set A converges uniformly to a complex-valued function f if for all ɛ > 0, there exists some N N such that for all x A, f n (x) f(x) < ɛ. 2.4.1 Lemma The complex sequence (f n ) converges uniformly to f iff sup f n (x) f(x) approaches 0. x
7 2.4.2 Lemma Given a complex sequence (f n ) converging uniformly to f, if each f n is continuous, then so is f. 2.5 Theorem A complex valued function is continuous if for all open sets in C, the preimage of open sets is open. 2.6 Theorem If K C is compact iff it is closed and bounded. 2.6.1 Corollary If K C and f : C C is continuous, then f(k) is closed and bounded. 2.6.2 Corollary If K C and f : C C is continuous, then f attains its maximum and minimum at points in K. 2.7 Definition: Partial Sum We define the partial sum of the sum of complex numbers a n is given by S m = a 0 + a 1 + + a m 2.7.1 Theorem: The n th Term Test If a n converges, then lim n a n = 0 2.7.2 Theorem: The Weierstrass Comparison Test If φ n (z) b n for all z A C and for all n N with b n converging, then converges uniformly. 2.8 Definition: Power Series We define the power series centres at z 0 to be complex numbers. φ n (z) a n (z z 0 ) n, where a n is a sequence of
8 2.9 Definition: Radius Of Convergence We define the raduis of convergence of a power series by: { } R = sup z a n z n converges z C 2.9.1 Abel s Lemma If a N z n os convergent at z = w 0, then ot converges absolutely at every z with Z < w 0, and it converges uniformly on every disk r (0) with r < w 0. 2.9.2 Corollary If R is the radius of convergence of a n z n, then the sum: converges absolutely at every z with z < R converges uniformly on the disk r (0) for all r < R diverges at every z with z > R.
Chapter 3 Holomorphic Functions 3.1 Definition: Differentiability We say that a function f : C A C with A open is (C) differentiable at z 0 A if the following limit exists: f f(z) f(z 0 ) (z 0 ) = lim z z0 z z 0 3.1.1 Lemma: Properties If f and g are differentiable at z 0, then we have the following properties: f + g is differentiable, and (f + g) (z 0 ) = f (z 0 ) + g (z 0 ) cf is differentiable for all c Z and (cf) (z 0 ) = cf (z 0 ) fg is differentiable, and (fg) (z 0 ) = f (z 0 )g(z 0 ) + g (z 0 )f(z 0 ) ) (z 0 ) = f (z 0 )g(z 0 ) g (z 0 )f(z 0 ) If g(z 0 ) 0 then f g is differentiable, and ( f g (g(z 0 )) 2 If f is differentiable at z 0 and g is differentiable at w 0 = f(z 0 ), then (g f) is differentiable at z 0, and (g f) (z 0 ) = g (f(z 0 ))f (z 0 ) If f is differentiable at z 0, w 0 = f(z 0 ) and f 1 is continuous at w 0, then f 1 is differentiable at w 0, and (f 1 ) (w 0 ) = 1 f (z 0 ) 3.1.2 Corollary Let f = f(z) f(z 0 ) and z = z z 0. f has a derivative iff r(z) where lim z 0 z = 0 f = f (z 0 ) z + R(z)
10 3.2 Definition: Holomorphic A function f is called holomorphic at a point z 0 if it is differentiable at all points in an open neighbourhood of z 0. f is called holomorphic in an open set Ω if it is differentiable at all points in Ω. 3.2.1 Lemma If f = u+iv is differentiable at z 0, then it satisfies the Cauchy Riemann equations, namely: u y = v x u x = v y or equivalently f y = i f x 3.2.2 Lemma If f is R differentiable at z 0 and it satisfies the Cauchy Riemann equations, then it is differentiable at z 0. 3.2.3 Corollary Suppose that f x and f y exist, are continuous Ω with Ω open and satisfy the Caushy Riemann equations. Then f is holomorphic in Ω.
Chapter 4 Integration 4.1 Definition: Path A path in C is any continuous map γ : [a, b] C. A reparamatrisation of a path γ : [a, b] C is a composition γ φ where φ : [ã, b] [a, b], where φ and φ 1 are continuous, and φ(ã) = a and φ( b) = b. 4.2 Definition: Path Integration Xe define the integral of a continuous function f along a piecewise smooth (i.e. continuous and differentiable) path γ by: b f(z) dz = f(γ(t))γ (t) dt 4.2.1 Lemma γ f(z) + g(z) dz = γ f(z) dz + γ g(z) dz γ Path integrals are independent of reparametrisation of the path. 4.3 Definition: Length The length of a piecewise smooth path γ is given by: 4.4 Theorem l(γ) = a b a γ (t) dt If γ is a path in an open set Ω and f is a function in Ω with a complex anti-derivative F, then: f(z) dz = F (γ(b)) = F (γ(a)) 4.5 Theorem: Cauchy s Theorem γ The path integral of holomorphic functions along a closed path is 0 under some assumptions.
12 4.6 Theorem: Cauchy-Goursat s Theorem If f is holomorphic in Ω, and open set in the complex plane, and is a closed triangle in Ω, then: f(z) dz = 0 δ where δ is the oriented boundary of. 4.6.1 Theorem: Extension of Cauchy s Theorem Cauchy s Theorem is also true for holomorphic functions defined on Ω along the boundaries of the following regions: Triangles with Ω Polygonal Regions D, (i.e. regions which are the disjoint union of simple, non-selfintersecting, closed, piecewise linear paths) with D Ω 4.7 Definition: Residue The residue of a function f on a region D at a point a is given by: Res a f = 1 f(z) dz 2πi where ɛ is chosed such that ɛ (a) D \ {a}. δ ɛ(a) 4.7.1 Theorem: The Residue Theorem If D is a bounded, simple region, a 1, a 2,..., a n D and f is holomorphic in a neighbourhood of D, then n f(z)z dz = 2πi Res ak f δ where ɛ(a k ) = ɛ \ {a k }, and where a 1, a 2,..., a k are points of singularity of f. 4.7.2 Lemma If g is holomorphic in ɛ (a) \ {a} and g is continuous on ɛ (a) with g(a) = 0, then Res a g(z) z a = 0. k=1 4.8 Theorem: Cauchy s Integral Formula If f is holomorphic in Ω and D Ω is a bounded, simple region, then: f(z) = 1 f(ζ) 2πi ζ z dζ δd
13 4.9 Theorem If f is holomorphic in a disk r (a), then f has an expansion: where c n = 1 2πi f(z) = f(ζ) dζ (ζ a) n+1 c n (z a) n r (a)