Complex Analysis. Travis Dirle. December 4, 2016

Transcription

1 Complex Analysis

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3 Complex Analysis Travis Dirle December 4, 2016

4 2

5 Contents 1 Complex Numbers and Functions 1 2 Power Series 3 3 Analytic Functions 7 4 Logarithms and Branches 13 5 Complex Integration 15 6 Cauchy s Theorem and Its Consequences 17 7 Isolated Singularities 21 i

6 CONTENTS ii

7 Chapter 1 Complex Numbers and Functions Basic Properties: A complex number takes the form z = x + iy where x and y are real and i is an imaginary number that satisfies i 2 = 1. We call x and y the real part and the imaginary part of z, respectively, and write x = Re(z) and y = Im(z). The complex numbers can be visualized as the usual Euclidean plane by the following: the complex number z = x + iy C is identified with the point (x, y) R 2. Naturally, the x and y axis of R 2 are called the real axis and imaginary axis. The natural rules for adding and multiplying complex numbers: if z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2, then and also z 1 + z 2 = (x 1 + x 2 ) + i(y 1 + y 2 ) z 1 z 2 = (x 1 x 2 y 1 y 2 ) + i(x 1 y 2 + y 1 x 2 ). The notion of length, or absolute value of a complex number is identical to the notion of Euclidean length in R 2. We define the absolute value or modulus of a complex number z = x + iy by z = (x 2 + y 2 ) 1/2 so that z is precisely the distance from the origin to the point (x, y). The complex conjugate of z = x + iy is defined by Also, z = x iy z ± w = z ± w, zw = z w, z/w = z/ w, z + w z + w. 1

8 CHAPTER 1. COMPLEX NUMBERS AND FUNCTIONS Also we have that We also have that Re(z) = z + z 2 and Im(z) = z z 2i z 2 = zz and as a consequence 1 z = z z 2 Any non-zero complex number z can be written in polar form z = re iθ where r > 0; also θ R is called the argument of z (defined uniquely up to a multiple of 2π) and is denoted by arg z, (we normalize arg by insisting that arg z ( π, π] and denote this Arg z) and e iθ = cos θ + i sin θ We have that r = z, and θ is simply the angle (with positive counterclockwise orientation) between the positive real axis and the half line starting at the origin and passing through z. We also see that e z = e Re z and arg(e z ) = Im(z). Finally, note that if z = re iθ and w = se iφ, then zw = rse i(θ+φ) A connected open set in C will be called a domain. Lemma (de Moivre s Formula) (cos θ + i sin θ) n = cos(nθ) + i sin(nθ). Theorem Suppose that z is a non-zero complex number and that n is a positive integer. then z has exactly n distinct complext nth-roots. These roots are given in polar form by [ ( ) ( )] n Arg z + 2kπ Arg z + 2kπ z cos + i sin n n for k = 0, 1,..., n 1. 2

9 Chapter 2 Power Series Theorem A series of complex numbers n=1 z n converges if and only if given each ɛ > 0 there is an index N such that m k=n z k < ɛ holds whenever m n N. Definition If the series n=1 z n converges, then the original series n=1 z n is said to be absolutely convergent. We then have that z n n=1 z n. An important characteristic of an absolutely convergent series is that the terms of such a series can be permuted in any arbitrary fashion without influencing the convergence or the value of the sum. Definition A series that converges, but not absolutely, is called a conditionally convergent series. If z n = x n + iy n, the complex series absolutely) if and only if both real series n=1 z n converges (resp. converges n=1 x n and n=1 y n converge (resp. converge absolutely). We have that z n = x n + i y n. n=1 n=1 Lemma A doubly infinite series of complex numbers n= z n converges if and only if both the series n=0 z n and n=1 z n converge, in which event z n = z n + z n. n= n=1 The notion of absolute convergence carries over to doubly infinite series, with the expected results: if n= z n converges, then n= z n converges; moreover, the terms of the latter series can be arbitrarily rearanged without affecting either the convergence or the value of the sum. 3 n=1 n=1 n=0

10 CHAPTER 2. POWER SERIES Definition Let (f n ) be a sequence of complex valued functions each of which are defined in an open set U of C. Let s n be the nth partial sum: s n = f 1 + f f n If the sequence of functions (s n ) converges pointwise in U to the limit function f, then we write f = n=1 f n and say that the infinite series is pointwise convergent in U with sum f. If (s n ) converges uniformly on a subset A of U, then the infinite series is uniformly convergent on A. Finally, if (s n ) converges uniformly on each compact set in U, then the infinite series is termed normally convergent in U. In order to certify that a series is normally convergent in U, we need only check that it is uniformly convergent on each closed disk in U. We speak of n=1 f n as absolutely convergent in U if n=1 f n is pointwise convergent in U. Theorem (Weierstrass M-test) Suppose that each term in a function series n=1 f n is defined on a set A. If there exists a sequence (M n ) of real numbers such that the estimate f n (z) M n holds for every z in A and such that the series n=1 M n converges, then n=1 f n converges absolutely and uniformly on A. Definition Let S be a set, and f a bounded function on S. Then we define the sup norm f S = f = sup f(z), z S Definition We say that {f n } is a Cauchy sequence, if given ɛ, there exists N such that if m, n N, then f n f m < ɛ. Theorem If a sequence {f n } of functions on S is Cauchy, then it converges uniformly. Recall from calculus: Theorem (The Ratio Test) Let a n be a series with positive terms and suppose that a n+1 lim = ρ n a n Then (i) the series converges if ρ < 1, (ii) the series diverges if ρ > 1 or ρ is infinite, (iii) inconclusive if ρ = 1. Theorem (The Root Test) Let a n be a series with a n 0 for n N, and suppose that n an = ρ 4 lim n

11 CHAPTER 2. POWER SERIES Then (i) the series converges if ρ < 1, (ii) the series diverges if ρ > 1 or ρ is infinite, (iii) inconclusive if ρ = 1. Theorem Let a n z n be a power series. If it does not converge absolutely for all z, then there exists a number r such that the series converges absolutely for z < r and does not converge absolutely for z > r. The number r is called the radius of convergence of the power series. If the power series converges absolutely for all z, then we say that its radius of convergence is infinity. When r is 0, then the series converges absolutely only for z = 0. If r is non-zero, then the power series is called a convergent power series. Definition Suppose that z 0 C. We refer to a function series of the type a n (z z 0 ) n = a 0 + a 1 (z z 0 ) + a 2 (z z 0 ) 2 +, n=0 where a 0, a 1, is a sequence of complex numbers, as a Taylor/power series centered at z 0. Definition With any such Taylor series we associate an extended real number ρ by the rule ( 1 n ρ = lim sup an ) n Here we observe the conventions 1/0 = and 1/ = 0. The quantity ρ is known as the radius of convergence of the series. When ρ > 0 the open disk (z 0, ρ) is called the disk of convergence. 5

12 CHAPTER 2. POWER SERIES 6

13 Chapter 3 Analytic Functions Definition Let f be a function defined in some neighborhood of a point z 0. We say that f is analytic at z 0 if there exists a power series a n (z z 0 ) n n=0 and some r > 0 such that the series converges absolutely for z z 0 < r, and such that for such z, we have f(z) = a n (z z 0 ) n. n=0 Suppose f is a function on an open set U. We say that f is analytic on U if f is analytic at every point of U. Theorem A taylor series diverges for any z satisfying z z 0 > ρ. If ρ > 0, the series converges absolutely and normally in the disk (z 0, ρ), so the function f defined by f(z) = n=0 a n(z z 0 ) n is analytic in (z 0, ρ). The coefficient a n is then related to f through the formula a n = f (n) (z 0 ). n! If S is an arbitrary set, not necessarily open, then a function is analytic on S if it is the restriction of an analytic function on an open set containing S. Theorem Suppose f is analytic in an open set U, that z 0 U, and that (z 0, r) U. Then f can be represented in as a Taylor series centered at z 0. This expansion is uniquely determined by f: if f(z) = n=0 a n(z z 0 ) n in, then the coefficient a n is given by a n = f (n) (z 0 )/n!. Theorem The power series f(z) = n=0 a n(z z 0 ) n defines an analytic function in its disc of convergence. The derivative of f is also a power series 7

14 CHAPTER 3. ANALYTIC FUNCTIONS obtained by differentiating term by term the series for f, that is, f (z) = na n (z z 0 ) n 1 n=0 Moreover, f has the same radius of convergence as f. Corollary A power series is infinitely complex differentiable in its disc of convergence. Definition By a Laurent series centered at z 0 we mean a doubly infinite function series of the form a n (z z 0 ) n n= = + a 2 (z z 0 ) 2 + a 1 z z 0 + a 0 + a 1 (z z 0 ) + a 2 (z z 0 ) 2 +, where each a n is a complex constant. Definition We assign to any such Laurent series two non-negative extended real numbers ρ O and ρ I, its outer and inner radii of convergence, via the formulas ρ O = ( lim sup n ) 1 n an, ρ I = lim sup n n a n When ρ I < ρ O, D = {z : ρ I < z z 0 < ρ O } is called the ring/annulus of convergence of the series. Theorem The Laurent series diverges for any z satisfying z z 0 > ρ O or z z 0 < ρ I. If ρ O > 0, the series n=0 a n(z z 0 ) n converges absolutely and normally in the disk (z 0, ρ O ), so f O (z) = n=0 a n(z z 0 ) n defines a function that is analytic in. If ρ I <, the series n=1 a 1(z z 0 ) n converges absolutely and normally in the open set D I = {z : z z 0 > ρ I }, so f I (z) = n=1 a n(z z 0 ) n defines a function that is analytic in D I. If ρ I < ρ O, the full Laurent series converges absolutely and normally in the set D = {z : ρ I < z z 0 < ρ O }, so that the function defined by f(z) = n= a n(z z 0 ) n = f I (z) + f O (z) is analytic in D. The coefficient a n is then related to f through the formula a n = 1 f(z)dz 2πi z z 0 =r (z z 0 ) n+1 for any number r (ρ I, ρ O ). Theorem Suppose that a function is analytic in an annulus centered at z 0, then f can be represented in the annulus as a Laurent series centered at z 0 with the coefficient a n as given above. 8

15 CHAPTER 3. ANALYTIC FUNCTIONS Definition Suppose that U is a non-empty open subset of the complex plane, that f is a function whose domain contains U, and that f is (complex) differentiable at every point of U, then f is analytic/holomorphic in U. A function whose domain is an open set in which that function is analytic is known as an analytic function. A function is analytic at a point z 0 if f is differentiable at every point in a neighborhood of z 0. Suppose that a function f(x + iy) = u(x, y) + iv(x, y) is differentiable at z 0 = x 0 + iy 0. By definition this requires that z 0 be interior to the domain-set of f and that f (z 0 ) = lim z z0 (f(z) f(z 0 ))/(z z 0 ) exist. Note that there is no constraint to the manner in which z tends to z 0. We end up getting that f (z 0 ) = u x (z 0 ) + iv x (z 0 ) = f x (z 0 ) as well as f (z 0 ) = v y (z 0 ) iu y (z 0 ) = if y (z 0 ). From these we get: The Cauchy-Riemann Equations: u x = v y u and y = v x A necessary condition for a function f = u + iv to be differentiable at a point z 0 is that u and v satisfy the Cauchy-Riemann equations at z 0. Theorem Suppose that a function f = u + iv is defined in an open subset U of the complex plane and that the partial derivatives u x, u y, v x, and v y exist everywhere in U. If each of these partial derivatives is continuous at a point z 0 of U and if the Cauchy-Riemann equations are satisfied at z 0, then f is differentiable there and f (z 0 ) = f x (z 0 ) = if y (z 0 ). Definition A real-valued function u(x, y) which is twice continuously differentiable and satisfies Laplace s equation u xx + u yy = 0 throughout a domain D is said to be harmonic in D. Theorem If f = u + iv is analytic in a domain D, then u and v are harmonic there. Lemma Suppose that u is a real-valued function which is defined in a plane domain D and that u x (z) = u y (z) = 0 for every z D. Then u is constant in D. Theorem Suppose that a function f is analytic in a domain D and that f (z) = 0 for every z D. Then f is constant on D. Theorem Let f = u + iv be analytic in a domain D. If any one of the functions u, v, or f is constant in this domain, then f itself is constant in D. Theorem If f, g are analytic on U, so are f + g, fg. Also f/g is analytic on the open subset of z U such that g(z) 0. If g : U V is analytic and f : V C is analytic, then f g is analytic. 9

16 CHAPTER 3. ANALYTIC FUNCTIONS Theorem If a function f is analytic in a domain D and if there exists a point ζ 0 of D with the property that f (n) (ζ 0 ) = 0 for every positive integer n, then f is constant in D. Theorem Suppose that a function f is analytic and non-constant in a domain D and that z 0 is a point of D for which f(z 0 ) = 0. Then f can be uniquely represented in D in the fashion f(z) = (z z 0 ) m g(z), where m is a positive integer and g : D C is an analytic function that obeys the condition g(z 0 ) 0. Corollary Suppose that a function f is analytic and non-constant in a domain D and that z 0 is a point of D. Then f can be uniquely represented in D in the fashion f(z) = f(z 0 ) + (z z 0 ) m g(z) where m is a positive integer and g : D C is an analytic function that obeys the condition g(z 0 ) 0. The integer m is called the multiplicity/order of f at z 0. We say that f takes the value f(z 0 ) with order m at z 0. The multiplicity of f at z 0 is the smallest positive integer m for which f (m) (z 0 ) 0. Theorem (L Hospital s Rule) Let f and g be functions that are analytic and non-constant in (z 0, r). Assume that each of these functions has a zero at the point z 0. Then f(z) lim z z 0 g(z) = lim f (z) z z 0 g (z), understood to mean that either both limits exist and are the same, or else neither limit exists. Definition A function that is analytic on the whole complex plane is called entire. Definition Let U be an open set. A subset E of U is termed a discrete subset of U if E has no limit point that belongs to U. So E must be a set of isolated points. We speak of f as a discrete mapping of U if for each fixed complex number w, the set E w = {z U : f(z) = w} is a discrete subset of U. Theorem (Discrete Mapping Theorem) If a function f is analytic and non-constant in a domain D, then f is a discrete mapping of D. Note that the set of zeros of a function in a domain D where that function is analytic and non-constant, if non-empty, consists entirely of isolated points. Theorem (Principle of Analytic Continuation) If functions f and g are analytic in a domain D and if f(z) = g(z) for all z belonging to some subset A of D that has a limit point in D, then f(z) = g(z) for every z in D. 10

17 CHAPTER 3. ANALYTIC FUNCTIONS Theorem If functions f and g are analytic in a domain D and if f(z)g(z) = 0 for every z D, then either f(z) = 0 for every z D or g(z) = 0 for every z D. Some important functions: cos z = n=0 ( 1) n z 2n, and sin z = (2n)! n=0 ( 1) n z 2n+1 (2n + 1)! cos z = eiz + e iz, and sin z = eiz e iz 2 2i e z z n = n! n=0 11

18 CHAPTER 3. ANALYTIC FUNCTIONS 12

19 Chapter 4 Logarithms and Branches Definition For z 0 we define the logarithm of z, log z to be the multivalued function log z = log z + i arg z = log z + iarg z + 2πim for m = 0, ±1, ±2,.... The values of log z are precisely the complex numbers w such that e w = z. We define the principle logarithm denoted Log z to be Log z = log z + iarg z, z 0. In general, we have that log(z 1 z 2 ) = log z 1 + log z 2, but this isn t to be expected of the principle logarithm. Definition For any logarithm w of z, the complex number e λw is called the λ-power of z associated with w. The choice w = Log z gives rise to the principle λ-power of z. z λ = e λlogz Definition If f : U C is an analytic function and if D is a domain containted in f(u), then by a branch of f 1 in D, we mean a continuous function g : D U that satisfies the condition f(g(z)) = z for all z D. Theorem Suppose that f : U C is an analytic function and that g is a branch of f 1 in a domain D. Let z 0 be a point of D. If f (g(z 0 )) 0, then g is differentiable at z 0 and g (z 0 ) = 1/f (g(z 0 )). Consequently, if f is free of zeros in g(d), then g is analytic in D, where its derivative satisfies g (z) = 1/f (g(z)). Definition If p 2 is an integer and if D is a domain, then by a branch of the pth-root function in D, we mean an analytic function g : D C with the feature that (g(z)) p = z for all z D. Definition A branch of the logarithm function in a domain D is an analytic function L : D C with the property that e L(z) = z for all z D. Moreover L 1 (z) = f (L(z)) = 1 e = 1 L(z) z If a branch of L of log z exists in a domain D, then D cannot contain the origin for z = e L(z) 0. 13

20 CHAPTER 4. LOGARITHMS AND BRANCHES Theorem Suppose that a branch L of the logarithm function exists in a domain D. Then the collection of all branches of the logarithm function in D consists of the functions L + 2kπi, where k is an integer. Definition If L is a branch of the logarithm function in a domain D and if λ is a complex number, then the branch of the λ-power function in D associated with L is the function h λ : D C defined by h λ (z) = e λl(z) Definition If λ = 1/p for p 2, then (h 1/p (z)) p = (e L(z)/p ) p = e L(z) = z making h 1/p a branch of the pth-root function in D the branch of the pth-root function in D associated with L. Theorem Suppose that a function f is analytic and free of zeros in a domain D. There exists a branch of log f(z) in D if and only if f (z)dz = 0 f(z) for every closed, piecewise smooth path in D. If g is a branch of log f(z) in D, then the collection of all such branches consists of the functions g + 2kπi, where k is an integer. Theorem Suppose that U is an open set, that f : U C is an analytic function, and that g is a branch of f 1 in a domain D. Then g is an analytic function. Theorem Let f(z) = c(z z 1 ) m 1 (z z 2 ) m2 (z z r ) mr, where c, z 1, z 2,..., z r are complex numbers satisfying c 0 and z j z k for j k, and where m 1, m 2,..., m r are non-zero integers. There exists a branch of log f(z) in a domain D if and only if m 1 n(, z 1 ) + m 2 n(, z 2 ) + + m r n(, z r ) = 0 is true for every closed, piecewise smooth path in D 14

21 Chapter 5 Complex Integration Definition By a path in the complex plane we mean a continuous function of the type : [a, b] C. The range of a path is called its trajectory, denoted. The initial and terminal points of the path are the points (a) and (b), respectively. When these values coincide, we call a closed path. If (t) (s) for t s with the possible exception that (a) = (b), we call the path simple. A Jordan curve is a trajectory of a simple, closed path. A path given by (t) = x(t) + iy(t) for a t b is termed a smooth path if its derivative (t) = ẋ(t) + iẏ(t), with respect to the real parameter t, exists for each t [a, b] and if the function is continuous on [a, b]. Definition For a continuous function g : [a, b] C with g = u + iv, then we have b a g(t) dt = b a u(t) dt + i b a v(t). The Second Fundamental Theorem of Calculus also remains valid: if g, G : [a, b] C are continuous functions and Ġ(t) = g(t) for every t (a, b), then b a g(t) dt = [G(t)] b a = G(b) G(a). Definition Suppose (defined above) is a smooth path and that f is a complex function which is defined and continuous on the trajectory of. Then the complex line integral or contour integral of f along is f(z) dz = b a f[(t)] (t) dt Definition The integral of f along with respect to arclength is given by f(z) dz = b a 15 f[(t)] (t) dt

22 CHAPTER 5. COMPLEX INTEGRATION Definition The length l() of the path is given by b ẋ(t)2 dz = + ẏ(t) 2 dt a Lemma Suppose that f : A C and g : A C are continuous functions and that and β are piecewise smooth paths in A. (i) [f(z) + g(z)] dz = f(z) dz + g(z) dz; (ii) cf(z) dz = c (iii) if + β is defined, then +β f(z) dz for any constant c; f(z) dz = f(z) dz + β f(z) dz; (iv) if β is obtainable from by a piecewise smooth change of parameter, then (v) (vi) f(z) dz = β f(z) dz = f(z) dz f(z) dz; f(z) dz; f(z) dz Definition Suppose that U is an open set in C and that f is a function whose domain includes U. A function F : U C is a primitive for f in U if F is analytic in U and has F (z) = f(z) for every z in that set. Theorem Suppose that a function f is continuous in an open set U and that F is a primitive for f in U. If : [a, b] U is a piecewise smooth path, then f(z) dz = [F (z)] (b) (a) In particular, under the above hypothesis it is true that f(z) dz = 0 for every closed, piecewise smooth path in U. 16

23 Chapter 6 Cauchy s Theorem and Its Consequences Theorem Suppose that a function f is continuous in a plane domain D and that f(z) dz = 0 for every closed, piecewise smooth path in D. Then f has a primitive in D. Theorem (Cauchy s Theorem - Local Form) Suppose that is an open disc in the complex plane and that f is a function which is analytic in (or, more generally, is continuous in and analytic in \{z 0 } for some point z 0 of in ). Then f(z) dz = 0 for every closed, piecewise smooth path in. Definition Suppose that is a closed and piecewise smooth path and that z is a point of C\. The winding number n(, z) or index, of about z is defined to be: n(, z) = 1 2πi dζ ζ z. The winding number is an integer, it records the net number of complete revolutions of a path about a point. Lemma Let be a closed, piecewise smooth path in the complex plane and let U = C\. Then: (i) n(, z) remains constant as z varies over any component of U; (ii) n(, z) = 0 for any z belonging to the unbounded component of U; (iii) when is simple, either n(, z) = 1 for every z in the bounded component of U or n(, z) = 1 for all such z. For a simple, closed, and piecewise smooth path, we say is positively oriented if n(, z) = 1 for every z in the inside of the path, and negatively oriented if n(, z) = 1 for all such z. Theorem (Cauchy s Integral Formula - Local Form) Suppose that a function f is analytic in an open disc and that is a closed, piecewise smooth path 17

24 CHAPTER 6. CAUCHY S THEOREM AND ITS CONSEQUENCES in. Then for every z \. n(, z)f(z) = 1 f(ζ)dζ 2πi ζ z Corollary If a function f is analytic in an open set U, then it can be differentiated arbitrarily often in U and all its derivatives f, f,..., f (k),... are analytic there. Theorem (Morera s Theorem) Let a function f be continuous in an open set U. Assume that f(z) dz = 0 for every closed rectangle R in U whose R sides are parallel to the coordinate axes. Then f is analytic in U. Theorem Suppose that a function f is continuous in an open set U and analytic in U\{z 0 } for some point z 0 of U. Then f is analytic in U. Theorem Suppose that a function f is analytic in an open disc (z 0, r) and that f(z) m holds throughout, where m is a constant. Then for each positive integer k, the estimate f (k) (z) k!mr (r z z 0 ) k+1 is valid for every z. In particular, f (k) (z 0 ) k!mr k. Theorem (Liouville s Theorem) The only bounded entire functions are the constant functions on C. Theorem (Fundamental Theorem of Algebra) Any polynomial function p(z) = a 0 + a 1 z + + a n z n of degree n 1 has a root in C. Theorem A polynomial function p(z) = a 0 + a 1 z + + a n z n of degree n 1 has a factorization p(z) = c(z z 1 )(z z 2 ) (z z n ), in which z 1, z 2,..., z n are the roots of p and c is a constant. Theorem (Mean Value Theorem) If f is analytic in D and z 0 D, then f(z 0 ) is equal to the mean value of f taken around the boundary of any disc centered at z 0 and contained in D. That is, when D(z 0, r) D. f(z 0 ) = 1 2π 2π 0 f(z 0 + re iθ ) dθ Theorem (Maximum-Modulus Principle) Let a function f be analytic in a domain D. Suppose that there exists a point z 0 of D with the property that f(z) f(z 0 ) for every z D. Then f is constant in D. Theorem (Minimum-Modulus Principle) If f is a non-constant analytic function in a domain D, then no point z D can be a relative minimum of f unless f(z) = 0. 18

25 CHAPTER 6. CAUCHY S THEOREM AND ITS CONSEQUENCES Corollary Let D be a bounded domain and let f : D C be a continuous function that is analytic in D. Then f(z) reaches its maximum at some point on the boundary of D. Lemma (Schwarz s Lemma) Suppose that a function f is analytic in (0, 1) and that it obeys the conditions f(0) = 0 and f(z) 1 for every z. Then f (0) 1 and f(z) z for every z. Furthermore, unless f happens to be a function of the type f(z) = cz in, where c is a constant of modulus one, it is actaully true that f (0) < 1 and that f(z) < z when 0 < z < 1. Theorem (Hadamard s Three-Lines Theorem) Let S be the set {z : 0 < Re z < 1}, and let f : S C be a bounded, continuous function that is analytic in S. Suppose that f(iy) m 0 and f(1 + iy) m 1 for all real y, where m 0 and m 1 are constants. Then for all real y, whenever 0 < x < 1. f(x + iy) m 1 x 0 m x 1 Definition By a cycle we mean a finite sequence of closed, piecewise smooth paths in C. We write σ = ( 1, 2,..., p ) if 1, 2,..., p are the paths that make up the cycle σ. Suppose that σ = ( 1, 2,..., p ) is a cycle in a set A and that f : A C is a continuous function. We have that f(z) dz = f(z) dz + f(z) dz + + f(z) dz 1 2 p σ In particular, for z C\ σ we define the winding number of σ about z by n(σ, z) = 1 dζ 2πi ζ z And also, n(σ, z) = n( 1, z) + n( 2, z) + + n( p, z). Definition Let U be an open set. A cycle σ in U is said to be homologous to zero in U if n(σ, z) = 0 for every z C\U. Two cycles σ 0 = ( 1, 2,..., p ) and σ 1 = (β 1, β 2,..., β q ) in U are pronounced homologous in U if the cycle σ = ( 1,..., p, β 1,..., β q ) is homologous to zero in that set, or equivalently, if n(σ 0, z) = n(σ 1, z) for every z C\U. Finally, two non-closed piecewise smooth paths λ 0 and λ 1 are homologous in U if they share the same initial and terminal points and if the closed path = λ 0 λ 1 is homologous to zero in U. Theorem (Cauchy s Theorem - Global) Let σ be a cycle in an open set U. Then f(z) dz = 0 for every function f that is analytic in U if and only if σ σ is homologous to zero in U. σ 19

26 CHAPTER 6. CAUCHY S THEOREM AND ITS CONSEQUENCES Corollary If a function f is analytic in an open set U and if σ 0 and σ 1 are cycles in U that are homologous in this set, then σ 0 f(z) dz = σ 1 f(z) dz. Corollary If a function f is analytic in an open set U and λ 0 and λ 1 are non-closed piecewise smooth paths in U that are homologous in this set, then λ 0 f(z) dz = λ 1 f(z) dz. Theorem (Cauchy s Integral Formula - Global) Suppose that a function f is analytic in an open set U and that σ is a cycle in U which is homologous to zero in this set. Then n(σ, z)f(z) = 1 f(ζ)dζ 2πi σ ζ z for every z U\ σ. Theorem Suppose that a function f is analtyic in an open set U, that k is a non-negative integer, and that σ is a cycle in U which is homologous to zero in this set. Then n(σ, z)f (k) (z) = k! f(ζ)dζ 2πi σ (ζ z) k+1 for every z U\ σ. Theorem (Goursat s Theorem) Let be a Jordan contour, and let D be the inside of with f : D C continous and also analytic in D. Then f(z) dz = 0 and f (k) (z) = k! f(ζ)dζ 2πi (ζ z) k+1 for every z D and every non-negative integer k. Theorem Let z be a point of the complex plane. If α and β are closed, piecewise smooth paths in C\{z} that are freely homotopic in C\{z}, then n(α, z) = n(β, z). Corollary Let U be an open set in the complex plane, and let α and β be closed, piecewise smooth paths in U. If α and β are freely homotopic in U, then they are homologous in this set. Definition A closed path in a set A is said to be contractible / null homotopic in A if is freely homotopic in that set to a constant path. Theorem Let U be an open set and let be a closed, piecewise smooth path in U. If is contractible in U, then is homologous to zero in this set. Definition A domain D is simply connected under the condition that every closed and piecewise smooth path in D, hence every cycle in D, is homologous to zero in that domain. Theorem Let D be a domain. Then D is simply connected if and only if every function that is analytic in D possesses a primitive in this domain. 20

27 Chapter 7 Isolated Singularities Definition We say that a function f has an isolated singularity at a point z 0 provided there exists an r > 0 with the property that f is analytic in the punctured disc (z 0, r), yet not analytic in the full open disc (z 0, r). This situation can come about for one of two reasons: either z 0 does not belong to the domain-set of f, or, z 0 is a member of the domain-set but is a point at which f is discontinuous. We call f analytic modulo isolated singularities in an open set U if there is a discrete subset E of U, the singular set of f in U, with the feature that f is analytic in the open set U\E, but has a singularity at each point of E. Recall that if f is analytic in (z 0, r) and continuous in (z 0, r) then f is actaully analytic in (z 0, r). Definition Assume f has an isolated singularity at z 0. Let (z 0, r) be the punctured disc in which f is analytic. We know that f can be represented in as a sum of a Laurent series centered at z 0. The singularity falls into one of three categories depending on the character of the Laurent expansion. We say f has a removable singularity at z 0 if a n = 0 for every negative index n; to have a pole at z 0 if a n 0 holds for at least one, but for at most finitely many negative values of n; and to have an essential singularity at z 0 if a n 0 is true for an infinite number of negative integers n. An isolated singularity of a function f at a point z 0 is removable if and only if f(z 0 ) can be defined, as to render f differentiable at z 0. Theorem (Riemann Extension Theorem) Let a function f have an isolated singularity at a point z 0. The singularity is removable if and only if f is bounded in some punctured disc centered at z 0. Theorem Let a function f have an isolated singularity at a point z 0. The singularity is removable if and only if lim z z0 f(z) exists. 21

28 CHAPTER 7. ISOLATED SINGULARITIES We say that f has a pole of order m at z 0 if we can write f as f(z) = a m (z z 0 ) m + + a 1 z z 0 + a n (z z 0 ) n, with a m 0. Multiplying both sides by (z z 0 ) m we obtain for all z n=0 (z z 0 ) m f(z) = a m + a m+1 (z z 0 ) + = a n m (z z 0 ) n. The last series is a Taylor series that converges at every point in. If we let g(z) = n=0 a n m(z z 0 ) n we see that it is analytic in and that g(z 0 ) = a m 0 and that f(z) = n=0 g(z) for every z (z z 0 ) m Theorem Let m be a positive integer. A function f that is analytic in a punctured disc (z 0, r) has a pole of order m at z 0 if and only if f can be represented in in the fashion f(z) = g(z) (z z 0 ) m, where g is a function that is analytic in (z 0, r) and obeys the condition g(z 0 ) 0. Definition The residue at z 0 of the function f (denoted Res(z 0, f)), is the coefficient a 1 in its Laurent expansion. Or it is the coefficient of (z z 0 ) m 1 in the Taylor expansion of g about z 0. Thus we have that Res(z 0, f) = 1 (m 1)! lim d m 1 z z 0 dz [(z z 0) m f(z)]. m 1 If an analytic function f has a zero of order m at z 0, then 1/f has a pole of order m at z 0. Also, if a function f has a pole of order m at z 0, then 1/f has a zero of order m there, in the sense that 1/f has a removable singularity at z 0 an that upon its removal, 1/f acquires a zero of order m at that point. Theorem Let a function f have an isolated singularity at a point z 0. The singularity is a pole if and only if lim z z0 f(z) =. Moreover, the singularity is a pole of order m if and only if m is the unique positive exponent for which lim z z0 z z 0 m f(z) is a positive real number. Theorem If neither of two functions f and g has worse than a pole at a point z 0, then none of the functions f, f + g, fg, and, unless g vanishes identically in some punctured disc centered at z 0, f/g has worse than a pole at z 0. 22

29 CHAPTER 7. ISOLATED SINGULARITIES Definition Let U be an open set. A function f is called meromorphic in U provided f has at no point of U worse than a pole. Theorem Let a function f have an isolated singularity at a point z 0. The singularity is essential if and only if lim z z0 f(z) fails to exist either in the strict sense or as an infinite limit. Theorem (Casorati-Weierstrass Theorem) If a function f is analytic in a punctured disc (z 0, r) and has an essential singularity at its center, then f( ) is dense in the complex plane, i.e., the set C\f( ) has no interior points. Theorem (Picard s Theorem) If a function f is analytic in a punctured disc (z 0, r) and has an essential singularity at its center, then the set C\f( ) contains at most one point. Theorem (Residue Theorem) Suppose that a function f is analytic modulo isolated singularities in an open set U, that E is the singular set of f in U, and that σ is a cycle in U\E which is homologous to zero in U. Then f(z) dz = 2πi n(σ, z)res(z, f). σ z E Theorem (Argument Principle) Assume that a function f is meromorphic in an open set U. Let be a Jordan contour in U such that the Jordan curve does not pass through any zero or pole of f and such that the inside D of is contained in U. Then 1 2πi f (z)dz f(z) = Z P, where Z and P indicate the number of zeros and the number of poles, respectively, that f has in D, multiplicity being taken into account. Theorem (Rouché s Theorem) If D is the domain inside the trajectory of a Jordan contour, if f and g are functions that are analytic in some open set which contains D, and if the inequality f(z) g(z) < f(z) + g(z) holds at every point z in D, then f and g have the same number of zeros in D, provided that zero-counts are made with due regard for multiplicity. A restatement of the theorem says that if g(z) < f(z) for all z D then f and f + g have the same number of roots in D. The classical Rouché s Theorem says that if f(z) g(z) < f(z) for all z D then f and g have the same number of roots in D. 23

30 CHAPTER 7. ISOLATED SINGULARITIES Theorem Let R be a rational function of x and y whose domain includes the circle K(0, 1). Then 2π 0 R(cos θ, sin θ) dθ = 2π p Res(z k, f), where f(z) = z 1 R[(z + z 1 )/2, (z z 1 )/2i] and z 1, z 1,..., z p are the poles of f in the disc (0, 1). Theorem If f(z) = (a 0 + a 1 z + + a n z n )/(b 0 + b 1 z + + b m z m ) is a rational function in which m n + 2 and in which the denominator has no real roots, then for c 0 f(x)e icx dx = 2πi k=1 p Res[z k, f(z)e icz ], where z 1, z 2,..., z p are the poles of f in the half-plane H = {z : Im z > 0}. Furthermore, if all the coefficients of f are real numbers, then { } p f(x) cos(cx) dx = Re 2πi Res(z k, f(z)e icz ) and f(x) sin(cx) dx = Im { k=1 2πi k=1 } p Res(z k, f(z)e icz ) k=1 If c > 0, then the conclusions are valid even if m = n