Complex Analysis Important Concepts

Complex Analysis Important Concepts Travis Askham April 1, 2012 Contents 1 Complex Differentiation 2 1.1 Definition and Characterization.............................. 2 1.2 Examples.......................................... 3 2 Integrals in the plane C 3 2.1 Cauchy s Theorem..................................... 3 2.2 Cauchy s Formula...................................... 4 2.3 Morera s Theorem..................................... 5 2.4 Liouville s Theorem and Consequences.......................... 5 3 Mapping Properties 6 3.1 The Identity Theorem................................... 6 3.2 Open Mapping Theorem.................................. 7 3.3 Maximum Modulus Principle............................... 7 3.4 Minimum Modulus Principle............................... 7 3.5 Schwarz s Lemma...................................... 8 3.6 Automorphisms of the Unit Disc............................. 8 3.7 Poles, etc........................................... 8 3.8 Casorati-Weierstrass.................................... 9 3.9 Little Picard........................................ 10 3.10 Big Picard.......................................... 10 3.11 Meromorphic Functions.................................. 10 3.12 Rational Functions..................................... 10 3.13 Mobius Transforms..................................... 11 1

4 The Residue Calculus and Consequences 11 4.1 Laurent Series........................................ 11 4.2 Winding Numbers..................................... 13 4.3 The Residue Theorem................................... 13 4.4 Argument Principle..................................... 14 4.5 Local Properties...................................... 15 4.6 Rouchés Theorem...................................... 15 5 Branches and Riemann Surfaces: Two Case Studies 15 6 Construction of Analytic Functions 15 6.1 Conformal Maps...................................... 15 6.2 Analytic Continuation................................... 16 6.3 Monodromy Theorem................................... 16 6.4 The Gamma Function................................... 16 6.5 Weierstrass Product Theorem............................... 16 6.6 Mittag-Leffler........................................ 16 7 Harmonic Functions 16 8 Contour Integral Applications 16 9 References 16 1 Complex Differentiation 1.1 Definition and Characterization A function f is complex-differentiable at a point a if it satisfies where f(z) = f(a) + l(z a) + r(z) lim z a r(z) z a = 0 for some l C. We write f (a) = l. We note that if we identify C with R 2, this definition implies that f is totally differentiable in the real sense and the action of its Jacobian is like multiplication by a complex number. In particular, we have for z = x + iy Suppose l = c + id. We see that f(x, y) = f(x a, y a ) + J(a)((x, y) T (x a, y a ) T ) + r(x, y) 2

So we have J(a)(x, y) T = l(x + iy) = (c + id)(x + iy) = (cx dy, dx + cy) ( ) c d J(a) = d c From the above, it is clear that f being complex differentiable and f being totally differentiable as a map in the real sense that satisfies the Cauchy-Riemann equations are equivalent. For l 0, we can write l = re iθ. Thus, the Jacobian of a real differentiable function can be thought of as a rotation-dilation. There is a nice, simple sufficient criterion for total differentiability from real analysis; if the partial derivatives exist and are continuous at each point, then f has a total derivative. We note that analyticity at a point requires a neighborhood about that point in which the function is complex differentiable (there are other definitions). A function is called holomorphic in D if it is analytic at each point in D. 1.2 Examples The fact that the Cauchy-Riemann equations must be satisfied implies that f(z) = z is nowhere complex differentiable. This is a simple example of a function which is continuous but nowhere complex differentiable. A complex differentiable function which takes only real values is a constant. This gives that f = sin z and f = Rz are not complex differentiable. The function f(z) = z z = x 2 + y 2 is complex-differentiable at zero but not analytic at zero (use the C-R equations) 2 Integrals in the plane C 2.1 Cauchy s Theorem The source of all good things. - Henry McKean If a function f is analytic in a simply connected domain D, then for any γ D we have f(z) dz = 0 γ 3

2.1.1 Goursat s Proof for a Rectangle Let γ enclose a rectangle R 0 in D and set I = γ f(z) dz. If we split the rectangle into four equal size rectangles, then we have I = I 1 +I 2 +I 3 +I 4 where I i is the integral around the perimeter of one of the subrectangles. This can be seen by drawing a picture and noting the cancellation of integrals going in opposite directions. We then have that I 4 I j for some j = 1, 2, 3, 4. We then define R 1 to be the rectangle corresponding to this j. We then choose nested rectangles R i by repeating this process on R i 1. Because the (closure) of the rectangles are compact sets, we have that they re infinite intersection is necessarily non-empty and must be equal to some point z 0 R i. Let I Ri be the integral of f around the perimeter of R i. We have by construction I 4 n I Rn. We note that l(γ n ) = 2 n l(γ) where l is the length of the curve and γ n is the perimeter of R n. Finally, we have that f(z) = f(z 0 ) + f (z 0 )(z z 0 ) + o(z z 0 ) the first two terms above obviously have an antiderivative and integrate to zero around any γ n. This gives Thus, we see that 1 f(z) dz = o(z z 0 ) dz = γ n γ n γ n 2 2 n l(γ)o(1) dz = l(γ)2 2 4 n o(1) so I = 0. I 4 n l(γ)2 2 4 n o(1) 2.1.2 Applications Virtually everything in section 2 of these notes is a corollary of this theorem. 2.2 Cauchy s Formula If f is analytic in a domain D containing γ and z 0 is inside of γ, then f (n) (z 0 ) = n! f(z) dz 2πi (z z 0 ) n+1 We note that the integrand above is analytic inside γ except at z 0. Cauchy s theorem then gives us that the integral about γ is equal to the integral about the circle z z 0 = ɛ for some ɛ sufficiently small. We can then prove the base case (the induction is rather straight-forward). We have γ f(z) f(z) dz = dz z z 0 z z 0 =ɛ z z 0 γ 4

f(z 0 ) + f (z 0 )(z z 0 ) + o(z z 0 ) = dz z z 0 =ɛ z z 0 1 = f(z 0 ) dz + f o(z z 0 ) (z 0 ) dz + dz z z 0 =ɛ z z 0 z z 0 =ɛ z z 0 =ɛ z z 0 o(z z 0 ) = f(z 0 )2πi + 0 + dz z z 0 f(z 0 )2πi z z 0 =ɛ In particular we have that f is infinitely differentiable. Further, one can show using Cauchy s formula that f is locally equal to its power series (Taylor expansion). This is an incredibly useful fact as we will see in the next theorem. 2.3 Morera s Theorem If f satisfies that γ f(z) dz = 0 for every curve γ in a connected (but not-necessarily simply connected) domain D, then f is analytic. This follows simply by constructing an antiderivative F of f, given as F (z) = z a dξ where a is some point in D and the integral is taken over any curve in D. This definition is consistent because of the assumption. Then, it is easy to show that F is analytic and F (z) = f(z). Then, the fact that F is locally equal to its power series shows that f is analytic because it s the derivative of F. We note that the assumptions can be weakened to assuming γ f dz = 0 on curves γ enclosing triangles contained in D. This actually characterizes f being analytic (f is analytic in D if and only if the integrals about triangles in D are zero). 2.4 Liouville s Theorem and Consequences If f is an entire bounded function, then f is constant. We can prove this using Cauchy s formula for the derivatives of f. Say f is bounded by M. For any R f (n) (0) = n! dξ 2πi ξ =R ξn+1 5

f (n) (0) n! 2π n! 2π n! 1 R n ξ =R ξ =R ξ n+1 dξ M dξ Rn+1 Thus, we have f (n) (0) = 0 for each n 1 and thus f is a constant. Similar results can be acquired in this manner for f a + b z n. Liouville s theorem has many consequences. We consider a few here. 2.4.1 Fundamental Theorem of Algebra The fundamental theorem of algebra states that a non-constant polynomial p of degree n has exactly n roots (counting multiplicity). We note the following fact. Suppose p has a root, p(z 0 ) = 0. We write p(z) = n 0 a jz j. Then p(z) = p(z) p(z 0 ) n = a j (z j z j 0 ) = 0 n a j (z z 0 )(z j 1 + z 0 z j 2 + z0z 2 j 3 + ) 0 = (z z 0 ) n a j (z j 1 + z 0 z j 2 + z0z 2 j 3 + ) 0 = (z z 0 )q(z) where q(z) is degree n 1. Thus, if we can prove that p has a zero, then the theorem follows by induction. Suppose p is a non-constant polynomial with no zeros. We note that 1/p(z) 0 as z. Thus, we choose an R > 0 so that 1/ p 1 for z > R. On the compact set z R, 1/p is continuous and thus attains its maximum, call this M. Then we have 1/p max(m, 1) and Liouville s theorem gives us that 1/p is constant and hence p is constant, a contradiction. Thus, p has a zero. 3 Mapping Properties 3.1 The Identity Theorem If f and g are analytic on a domain D then the following are equivalent: 6

f = g {z D : f(z) = g(z)} include an accumulation point There exists z 0 D such that f (n) (z 0 ) = g (n) (z 0 ) for all n N This follows because if the zeros of an analytic function have an accumulation point on a domain, then that function is identically zero on that domain. To prove this fact, one can show that the accumulation points of the zeros of f are an open set. Further, the set is relatively closed in D by the continuity of f, so the fact follows by connectedness of D. A corollary of this fact is that analytic continuations are unique. Further, if a function blows up at the boundary of the set, then there is no analytic continuation of that function that crosses the boundary of the set. 3.2 Open Mapping Theorem If f is a non-constant analytic function from C to C, we have that f(a) is open for open sets A. (Note that this does not hold for real analytic functions, e.g. sin x maps R to [ 1, 1]). To prove this, we show that each point f(b) in the image of f(a) has a neighborhood contained in f(a). WLOG we have 0 = b = f(b). We note that f(z) = z n (a n + a n+1 z + ) = z n h(z) for some n with a n non-zero. This gives that h(z) is non-zero in some neighborhood of zero. Thus, we define h 0 = h 1/n on that neighborhood and f 0 = zh 0. We then have that f 0 (0) 0 and so the impilicit function theorem implies that f 0 maps a neighborhood of zero to a neighborhood of zero. Then, because f = (f 0 ) n and the map z z n sends neighborhoods of zero to neighborhoods of zero, we have that f is an open map. 3.3 Maximum Modulus Principle An immediate consequence of the open mapping theorem is the maximum modulus principle. It states that if f is analytic in a domain D and f achieves its maximum on D, then f is constant. This is clear. If a D then the open mapping theorem says that f(a) is an interior point of f(d) and hence there is a point in D with larger modulus. 3.4 Minimum Modulus Principle If f is analytic on a domain D and f achieves its minimum in D then f = 0 at that point. This is easy to prove from the above. If not, 1/f is analytic and has a maximum on D, a contradiction. 7

3.5 Schwarz s Lemma For D the unit disc, let f : D C be analytic and have f 1 on D and have zero as a fixed point f(0) = 0. Then, on D we have f(z) z f (0) 1 Further, if there is a point a such that f(a) = a or if f (0) = 1, then f is a rotation around zero, f(z) = e iθ z for some θ. To prove this theorem, we note that f(0) = 0 gives us f(z) = zg(z) for some g analytic. This gives us that if z = r < 1 then g(z) 1/r. The maximum modulus principle then gives us that g(z) 1/r for z r. Passing to the limit as r 1, we get g(z) 1 so f(z) z and f (0) = g(0) 1. Further, if f(a) = a or f (0) = g(0) = 1 then g achieves its maximum and this thus a constant of modulus 1. 3.6 Automorphisms of the Unit Disc An automorphism is a bijective map from a set to itself which is analytic and has an analytic inverse. Schwarz s lemma gives us that an automorphism of the unit disc which fixes zero is given by ϕ(z) = ζz for some ζ = 1 (apply Schwarz s lemma to ϕ and ϕ 1 ). What does a general automorphism of the disc look like? a D given by ϕ a (z) = z a āz 1 is an automorphism of D which satisfies ϕ a (a) = 0, ϕ a (0) = a, and ϕ 1 a = ϕ a. 3.7 Poles, etc. We have that the map ϕ a for some We define a removable singularity of f to be a point a such that while f is not defined, it is possible to define a holomorphic extension of f near a. For instance, we set f(a) = lim z a f(z). A non-essential singularity is a point such that there exists m Z for which (z a) m f(z) has a removable singularity. An essential singularity is a point that does not satisfy this property. We define the order of a point a as follows: let k be the smallest integer such that (z a) k f(z) has a removable singularity and (z a) k f(z) 0. Then we set ord(f; a) = k. We have: positive order a is a zero of f zero order a is a removable singularity 8

negative order a is a pole We introduce a different characterization of the different types of singularities. An isolated singularity a is... removable f is bounded in a punctured neighborhood of a a pole lim z a f(z) = essential f approximates each value in C arbitrarily closely in any punctured neighborhood of a The last equivalence is proved in the following theorem. 3.8 Casorati-Weierstrass If f has an essential singularity at a and U r (a) is a punctured disc about a, then for any b C and ɛ > 0 there exists z U r (a) such that f(z) b ɛ. Proof. Suppose there is an ɛ > 0 and b such that f(z) b ɛ on U r (a). Then g = 1/(f(z) b) has a removable singularity, as it is bounded. We consider the analytic extension then g(z) = a k (z a) k 0 Further, we have that where b 0 0 so h(0) 0. This gives that g(z) = (z a) m b k (z a) k = (z a) m h(z) 0 f(z) b = 1 (z a) m h(z) 1 so we see that f has at most a pole at a, a contradiction. A simple application of this theorem is to the following problem: characterize all surjective entire functions. We note that f : { z < 1} to an open bounded set. The complement of this set is then not dense in C. In particular, f does not have an essential singularity at infinity (see meromorphic functions below for a more rigorous definition of this idea) and is thus a polynomial. By surjectivity, this polynomial must be degree 1. Thus we see f = az + b for a 0. 9

3.9 Little Picard The Little Picard theorem states that an entire, non-constant function assumes each value in C with at most one exception. The exception is necessary, consider e z which is never zero. 3.10 Big Picard The Big Picard theorem states that in any neighborhood of an essential singularity for a function, the function assumes each value in C with at most one exception. The exception is necessary, consider e 1/z which is never zero with an essential singularity at 0. 3.11 Meromorphic Functions We say that a function is meromorphic on a domain D if it is analytic aside from a discrete set of (isolated) poles. It is clear that we can represent f locally as a quotient of analytic functions (as each pole is isolated). However, it is also globally possible to represent f as a quotient of analytic functions. Weierstrass products give this result in the case D = C. An example of a meromorphic function with infinitely many poles is whose poles are the set Z. cot πz = cos πz sin πz We can also define what it means for a function to be meromorphic on C. If D C then f : D C is meromorphic if f is meromorphic in D C amd the function ˆf(z) = f(1/z) is meromorphic on the set ˆD = {z C : 1/z D}. We say that the singularity of f at infinity is the same type of singularity as ˆf at 0. To have an isolated singularity at infinity, f must have no other singularities in {z : z > C} for some C > 0. 3.12 Rational Functions The rational functions are examples of meromorphic functions. They are given as a quotient of two polynomials. We note that the meromorphic functions on all of C are precisely the rational functions. To see this, the fact that there aren t singularities other than the possible singularity at infinity outside of a certain disk gives that there are only finitely many singularities on C. Thus, for a finite set of poles a i we have that the negative part of the Laurent expansion near a i is a polynomial in 1/(z a i ). If we subtract all of these polynomials from f we have an analytic function. Further, this analytic function must be a polynomial because of the condition at infinity. 10

3.13 Mobius Transforms A rational function gives a bijective map for C C if and only if it is a Mobius transform, i.e. f(z) = az + b cz + d for ad bc 0. We can identify a Mobius transform with a matrix given by ( ) a b c It can be easily checked that composition of Mobius transforms is equivalent to multiplication of the corresponding matrices (we note that two matrices correspond to the same Mobius transform iff they differ by a non-zero scalar factor). It is then easy to see that the inverse of a Mobius transform is given by f(z) = d dz b cz + a which comes from the corresponding formula for 2 2 matrices. Some facts: The Mobius transforms are precisely the automorphisms of the Riemann sphere C Any non-identity Mobius transform has at least 1 and at most 2 fixed points (consider the fixed point equation az + b = z(cz + d)). A Mobius transform is determined by its values at 3 distinct points (consider the above fact applied to f g 1 where f(z i ) = g(z i ) for i = 1, 2, 3) 4 The Residue Calculus and Consequences 4.1 Laurent Series If f is analytic in an annulus {r < z a < R} centered at z = a then f can be written as f(z) = g(z) + h(1/z) where g is analytic in U R (a) and h is analytic in U 1/r (a). For a fixed annulus, this representation is unique. We prove existence for the case of an annulus about zero. First, we note that as a consequence of Cauchy s theorem, we have G(ξ) dξ = G(ξ) dξ ξ =ρ ξ =P 11

for r < ρ < P < R and G analytic. Further, the function { f(z) G(ξ) = ξ z for ξ z f (z) for ξ = z is analytic for z in the annulus A ρ,p. This gives us ξ =ρ 1 dξ f(z) ξ z ξ =ρ ξ z dξ = 2πif(z) = ξ =P ξ =P f(z) = 1 2πi dξ f(z) ξ z ξ z dξ ξ =P f(z) = g(z) + h(1/z) g(z) = 1 2πi h(z) = 1 2πi ξ =P ξ =ρ ξ =ρ ξ =P ξ z dξ 1 2πi ξ z dξ z 1 ξz dξ ξ z dξ 1 ξ z dξ ξ =ρ ξ z dξ As this is possible for any r < ρ < P < R and we can merge analytic functions, we have an analytic function on the whole annulus A r,r. More often, we write the Laurent representation for f on A r,r (a) as f(z) = and we note that the coefficients are given by a n = 1 2πi n= ξ a =ρ a n (z a) n dξ (ξ a) n+1 for r < ρ < R. We note that the Laurent decomposition gives us yet another way to classify singularities. We have that an isolated singularity a is... removable a n = 0 for all n < 0 a pole of order k N a k 0 and a n = 0 for n < k essential a n 0 for infinitely many n < 0 12

4.2 Winding Numbers The formula for a winding number gives us a way to compute how many times a curve loops around a certain point. One can check for simple cases that the formula aligns with our intuitive idea of what this number should be. We have χ(α, z) = 1 2πi α 1 ξ z dξ for the winding number. We would like this definition to satisfy many things. An important property is that χ(α, z) is integral for z / range α. We prove this in the case that α is smooth. Let G(t) = t α (s) 0 α(s) z ds and F (t) = (α(t) z)e G(t). We note that F (t) = 0. In particular, we have F (1) = F (0), i.e. So we see that χ(α, z) is integral as desired. 4.3 The Residue Theorem (α(1) z)e G(1) = (α(0) z)e G(0) (α(0) z)e G(1) = (α(0) z) e G(1) = 1 G(1) = k 2πi for k Z We first define a residue. If we have an isolated singularity at a point a, we consider the Laurent expansion of f about that point f = a n(z a) n. Then we define the residue of f at a as Res(f, a) = a 0 The residue theorem is as follows: if D C is an elementary domain, z 1,... z k are finitely many points in D and f : D\{z 1,..., z k } C is analytic, then for α : [a, b] D\{z 1,..., z k } a piecewise smooth, closed curve we have α dξ = 2πi k Res(f, z k )χ(α, z k ) j=1 To prove this fact, we consider the function g given by f minus the negative part of the Laurent series for each z j. Calculating Residues: here are some formulas for calculating residues Simple pole: 13

Pole of order k: Res(f, a) = lim z a (z a)f(z) Res(f, a) = ( d dz ) k 1 ( (z a) k f(z) ) (k 1)! 4.4 Argument Principle We note the following fact: if f has at most an isolated pole at a then Res( f ; a) = ord(f; a) f We prove this fact for a = 0 and k < 0. Let ord(f; a) = k, then z k f(z) has a removable singularity and z k f(z) 0. We then write Taking derivatives term-by-term, we get 1 f = a n z n = a n z n + a 0 + n=k n=k a n z n 1 Then, we have f = 2 n=k 1 a n+1 (n + 1)z n + (n + 1)a n+1 z n n=0 f f = z k f z k f = z k z k 2 n=k 1 a n+1(n + 1)z n + n=0 (n + 1)a n+1z n n=k a nz n = 1 n=k a nz n + a 0 + 1 a nz n = ka kz 1 + analytic a k + 1 a j+kz j and we see that the residue is indeed k. This fact gives us the argument principle: if α is a curve that surrounds each point of its interior once and f is a meromorphic function, then we have α f f dξ = 2πi(N P ) where N and P denote the number of zeros and poles within α counting multiplicities. The proof is an application of the residue theorem. 14

4.5 Local Properties Let f be a non-constant analytic function in a domain D C. Let a D be fixed and f(a) = b. Let n N is the order of f(z) b at z = a. Then there exists neighborhoods U(a) D and V (b) such that any w V with w b has exactly n preimages z 1,..., z n U, i.e. f(z i ) = w for 1 i n. This fact follows from the argument principle. In particular, we see that the only surjective functions are linear functions f = cz + d with c 0. 4.6 Rouché s Theorem Let f and g be analytic functions on an elementary domain D and α be a closed curve in D which surround each point in its interior exactly one time. If g < f on α, then the functions f, f + g have the same number of zeros on the interior, counting multiplicities. The proof for a finite number of zeros is simple. Let h s (z) = f(z) + sg(z) for 0 s 1. These functions have no zeros on the image of α. Then, the argument principle gives us a formula for the zeros of h s which depends continuously on s. Further, the number of zeros is integral and we see that the number of zeros of h s is constant in s, giving the result. 4.6.1 Examples The following are a few examples of Rouché s theorem in action. They are taken from a homework assignment in McKean s Complex Variables class at CIMS. For z 7 2z 5 + 6z 3 z + 1 = 0, how many solutions are inside z = 1? Solution: On the circle z = 1, we have that z 7 2z 5 z z 7 + 2 z 5 + z 4 < 5 6 z 3 1 6z 3 + 1 Then by Rouche s Theorem we have that z 7 2z 5 + 6z 3 z + 1 and 6z 3 + 1 have the same number of zeroes inside z = 1. Because the zeroes of 6z 3 + 1 are given by z = ( ) 1 1 3 6 We know that all three of its zeroes have z = ( ) 1 1 3 6 < 1. So z 7 2z 5 + 6z 3 z + 1 has three zeroes inside z = 1. 15

For z 4 6z + 3 = 0, how many solutions satisfy 1 z 2. Solution: On the circle z = 1, we have that z 4 z 4 = 1 < 3 6 z 3 3 6z So by Rouche s Theorem, we have that z 4 6z + 3 and 3 6z have the same number of zeroes inside z = 1. The zero of 3 6z is at z = 1/2 so each function has one zero in the unit disc. On the circle z = 2, we have that 3 6z 3 + 6 z 15 < 16 z 4 z 4 Again by Rouche s, we get that z 4 6z + 3 and z 4 have the same number of zeroes inside of z = 2. So z 4 6z + 3 has four zeroes inside of z = 2 and one zero inside of z = 1 so it has three zeroes in between the circles. 5 Branches and Riemann Surfaces: Two Case Studies 6 Construction of Analytic Functions 6.1 Conformal Maps 6.1.1 Definition A linear map is orientation preserving if det T > 0 and angle preserving if T x T y x, y = x y T x, T y We call angle and orientation preserving linear maps conformal. A totally differentiable (but not necessarily linear) map is called locally conformal if its Jacobian is conformal at each point. If the map is bijective in addition to locally conformal, it is called globally conformal. We see from the definitions, that a complex function f is locally conformal if and only if it is analytic and its derivative is non-zero everywhere. 16

6.1.2 Riemann Mapping Theorem 6.1.3 Schwartz Reflection 6.1.4 Schwartz Christoffel Maps 6.1.5 Examples and Uses 6.2 Analytic Continuation 6.3 Monodromy Theorem 6.4 The Gamma Function 6.5 Weierstrass Product Theorem 6.5.1 Examples 6.6 Mittag-Leffler 6.6.1 Examples 7 Harmonic Functions 8 Contour Integral Applications 9 References In creating these notes I used primarily the following references: Complex Analysis by Eberhard Freitag and Rolf Busam Complex Analysis by Lars Ahlfors My course notes for Complex Variables at CIMS taught by Henry McKean. The Courant Math Wiki: http://math.nyu.edu/student_resources/wwiki/index.php/ Main_Page 17