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Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 1 Review 1. Compute the cubic roots of the number 1 + i. 2. Identify R2 with C. Write down the complex function, z 7→ f (z), corresponding to the following map from R2 to R2 x 2x + 3y = . y x + 0.5y Write down the function u, v such that f = u + iv. 3. Prove the following statements. (a) A real linear map f : R2 → R2 is complex linear if and only if f (i) = if (1). (b) Let a, b, c, d ∈ R and x a b x T = . y c d y There exists a unique pair of complex numbers λ and µ such that T (z) = λz + µz̄, (c) The map T (z) = λz + µz̄ is complex linear if and only if µ = 0. By a region, we mean an open connected subset of C. Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 1 2 Example Sheet 1: Complex Differentiation 1. Let r be a non-zero real number, c a real number and k ∈ C, satisfying the relation |k|2 > cr. Prove that the equation r|z|2 − k̄z − kz̄ + c = 0. represents a circle. Determine its centre and radius. 2. Let c ∈ R and k ∈ C. Prove that k̄z + kz̄ + c = 0 represents a straight line. 3. let f : C \ {0} → C \ {0} be the inversion f (z) = z1 . Prove that f takes a circle to a circle or a line. 4. Suppose that f : U → C is complex differentiable where U is a region. Define Ū := {z : z̄ ∈ U }. Prove that g : Ū → C given by the formula g(z) = f (z̄) is complex differentiable. Write g 0 in terms of f . 5. Let log z := log |z| + i arg z. Use the Cauchy-Riemann equation to prove that the logarithm function log : C \ (−∞, 0] → C is holomorphic. 6. Let f : {a + ib : a > 0} → C \ (−∞, 0] be the function z 7→ z 2 . Prove that f −1 is holomorphic. 7. Determine at which points the following functions are differentiable: zRe(z), z̄ , |z|2 z z̄. 8. Find all holomorphic functions f whose real part is u(x, y) = 2xy + 2x. 9. Let us consider a function f in polar coordinate (r, θ). Let U be a domain on which f (r, θ) = ũ(r, θ) + iṽ(r, θ) is defined. Prove the polar form of the Cauchy-Riemann equation : 1 ∂r ũ = ∂θ ṽ, r 1 ∂r ṽ = − ∂θ ũ. r 10. If F 0 (z) = f (z) in a region D we say F is a primitive of f . Prove that f (z) = |z|2 does not have a primitive. (Hint: Use Cauchy-Riemann equation.) 3 Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 2 Example Sheet 2. Conformal Map. and Möbius Transforms 1. Let r be a positive number and a, b ∈ C. Prove that the equation determines a circle or a line. |z−a| |z−b| =r 2. Write down the Möbius transform that takes (0, i, 1) to (0, 1, −1). 3. Let b be a complex number with |b| < 1. Let f (z) = −z + b . −b̄z + 1 (a) Prove that f maps the unit disc D = {z : |z| < 1} to itself. (b) Prove that |f (z)| = 1 if |z| = 1 (c) Give a formula for f −1 . (d) Prove that f : D → D is bijective. 4. Denote by SL2 the family of special Möbius transforms az + b : a, b, c, d ∈ R, ad − bc = 1 . SL2 = cz + d Let H = {x + iy : y > 0} denote the upper half plane. Prove that (a) Each map from SL2 takes H to H. (b) For any two points z, w ∈ H there exist a map from SL2 taking z to w. 5. Compute the Stereographic distance between 2i and ∞. 6. Prove that if z ∈ C∗ corresponds to a point (X, Y, Z) in S 2 , then − z̄1 corresponds to its antipodal point. 7. Prove that if z1 , z2 , z3 , z4 are distinct points in C then they lie in a circleline if and only if the cross ration [z1 , z2 , z3 , z4 ] is a real number. Hint to (4b): First find a map taking z to i. Try d = 0. Solve the equation f (z0 ) = i and take care of both the real and the imaginary parts. Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 3 4 Example Sheet 3: Power Series and Contour Integral P k 1. If we know ∞ k=0 ak (z − i) converge for z = 4 and diverges for z = −8, what can you say about the convergence of the following power series? ∞ X ∞ X n an (1 + i) , n=0 n an 9 , ∞ X n=0 2. Prove that the power series function f (z) = disc D of convergence. (−1)n an 6n . n=0 P∞ 3. Find an analytic continuous of the power series R1 4. Compute 0 f (t) dt where f (t) = t + 2i. n=0 an z n is analytic in its P∞ n=0 (1 − z)n . 5. Write down a parametrisation for the triangle, with vertices the origin, R and R + iR, and for each side of the triangle. 6. Let γ be the upper half of the unit circle centred √ at the origin. Use the following two parametrisations z(t) = −t + i 1 − t2 , −1 < t < 1 and z̃(t) = cos(t) + i sin(t), t ∈ (0, π) of γ to compute the integral Z 1 dz. γ z (Please do not use any theorems) R 7. Compute |z|=1 z̄dz and conclude that f (z) = z̄ does not have a primitive on {z : |z| < 2}. 2 8. Let f (z) = e−z . Is f a holomorphic function? (a) For R > 0, prove that Z 1 f (z)dz ≤ . R [R,R+iR] (b) Prove that √ π f (z)dz = . 2 [0,R] Z lim R→∞ (c) Use Goursat’s theorem to prove that √ Z ∞ Z ∞ π 2 2 . cos(2t )dt = sin(2t )dt = 4 0 0 Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 4 5 Example Sheet 4. Primitives and Caychy’s Integral formula R 1. Calculate γ sin(z)dz where γ = {t + it : 0 ≤ t ≤ 1}, firstly by the definition, then by finding a primitive. 2. For each of the following functions, calculate the indicated integrals without using Cauchy’s theorems. Hint overleaf. R 2 (a) γi zez dz where γ1 = [1+i, i] and γ2 is the locus of |z−a|+|z−b| = r where a, b, ∈ C, r ∈ R and r > |a − b|. R (b) γ (z2z+1) dz if γ the locus of |z − 27| + |z − 26| = 2. 3. Let γ be a smooth curve with parameterization z : [a, b] → C and φ a holomorphic function. Denote by γ̃ the image of γ by φ : γ̃ = φ(γ). Prove that if f is a continuous function then Z Z f (z)dz = f ◦ φ(z) φ0 (z)dz. γ φ(γ) 4. If f is an even continuous function, i.e. f (z) = f (−z), prove that for any r > 0, Z f (z)dz = 0. |z|=r P∞ n 5. Let f (z) = n=0 an (z − z0 ) with radius of convergence R > 0. Let r ∈ (0, R). Prove that Z 1 f (z) dz. ak = 2πi |z−z0 |=r (z − z0 )(k+1) Hint overleaf. 6. Let z1 , z2 ∈ C and r1 , r2 be two real numbers such that r1 + |z1 − z2 | < r2 . (The circle |z − z1 | = r1 is in the interior of the larger circle |z − z2 | = r2 .) Suppose there exists R1 < 12 r1 and R2 > r2 such that f is holomorhpic in the region U = {|z − z1 | > R1 } ∩ {|z − z2 | < R2 }. Prove that Z Z f (z)dz = |z−z1 |=r1 in two ways. f (z)dz |z−z2 |=r2 Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 6 (1) Run a vertical and a horizontal lines through z1 giving us four closed simple curves γi , such that Z Z XZ f (z)dz = f (z)dz − f (z)dz. i |z−z2 |=r2 γi |z−z1 |=r1 For each γi there is a line segment sliding the larger disc into two regions, one of which contains γi and on which f is holomorphic. (2) Divide the region U into two simply connected regions with boundary curves Γ1 and Γ2 satisfying Z Z Z Z f (z)dz + f (z)dz = f (z)dz − f (z)dz. Γ1 7. Calculate |z| = 1. |z−z2 |=r2 Γ2 R 1 dz |z|=1 (z−a)(z−b) Alternative : calculate r 6= 3. where a, b ∈ C and a, b do not line on the circle 1 dz, γ (z−2)(z−3) R |z−z1 |=r1 γ is a circle |z − z0 | = r where r 6= 2 8. Use Cauchy’s formula to prove that, for C the circle |z| = 1 and a, b complex numbers with |a| > 1 and |b| < 1, Z 1 2πi dz = . 3 (b − a)3 C (z − a) (z − b) Hints. Q 2 (b). |z − a| + |z − b| = r, for r > |b − a|, is an ellipse with foci a and b. . Q4. Write down an explicit power series expansion f for (z−zf0(z) )(k+1) Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 5 7 Example Sheet 5: Cauchy’s formula, derivative formulas, applications 1. Prove the following extension to Louville’s theorem. If f is an entire function with limz→∞ f (z) = 0, then f is a constant. z 2. Compute the following Integrals. Z Z ez+1 1 dz, dz, 2 |z|=1 z |z|=2 z + 1 3. Compute R |z|=2 Z |z|=0.5 1 dz, z2 + 1 Z [− 12 , 21 ] z dz. z2 + 1 z n (1 − z)−m dz where n, m ∈ N . 4. Denote by γ1 the segment from 1 to r > 1 on the real axis, γ2 the arc from r to w = reiθ along the circle centred at 0 of radius r. Here θ ∈ (0, π2 ). Prove that Z 1 dz = ln w, γ1 ∪γ2 z Recall ln w = ln r + iθ. R 5. Let and a be positive numbers. Calculate R z1 dz where R is the rectangular with vertices at −a − i, a − i, −a + i, and a + i by the definition of curve integrals. (I am aware you know a quicker way to calculate this.) Prove that Z Z 1 1 = πi, lim = πi. lim →0 [a+i,−a+i] z →0 [−a−i,a−i] z 6. Suppose that f is holomorphic in a region containing {|z − z0 | < 2} with f (z0 ) = 0, f 0 (z0 ) 6= 0, f (z) 6= 0 for z 6= z0 . Use Cauchy’s theorem to prove that Z 1 2πi dz = 0 . f (z0 ) |z−z0 |=1 f (z) 7. Prove that the function 1 sin z has singularities at nπ where n ∈ Z. Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 6 8 Example sheet 6. 1. Prove that if an entire function f satisfies |f (z)| ≤ c|z|n for some natural number n, then f is a polynomial of degree at most n. 2. Prove that if f is a non-constant entire function then f (C) is dense in C. P 1 z z log n . 3. Denote ζ(z) = ∞ n=1 nz the Riemann ζ function. Recall that n = e P 1 • If Re(z0 ) > 1, prove N n=1 nz converges uniformly on a disc D(z0 , δ). • Prove that ζ is analytic on {z : Re(z) > 1}. 4. Let f be a holomorphic function on a deleted disc D0 (z0 , r) = {z : 0 < |z − z0 | < r}. Show that the following statements are equivalent : (a) ord(f ; z0 ) = n; (b) there exists positive numbers c1 , c2 and δ such that c1 |z − z0 |n ≤ |f (z)| ≤ c2 |z − z0 |n , 0 < |z − z0 | < δ. 5. Suppose that both f and g have singularity at z0 of finite orders. Prove that ord(f g; z0 ) = ord(f ; z0 ) + ord(g; z0 ), 1 ord( ; z0 ) = − ord(g; z0 ). g 6. Let f be a non-constant holomorphic function in a region U . Let V ⊂ U be a closed bounded subset of U . Then for any w ∈ C, {z ∈ V : f (z) = w} has a finite number of elements or f is a constant on V . P∞ P n 7. Given two power functions f (z) = ∞ n=0 bn (z− n=0 an (z−z0 ) and g(z) = n z0 ) on {|z − z0 | < R}, suppose that there exists a sequence of numbers {zk } such that f (zk ) = g(zk ) and limk→∞ zk = z0 . Prove by induction that an = bn for all n. P n 8. Suppose that f (z) = ∞ n=−∞ an (z − z0 ) for R1 < |z − z0 | < R2 . Prove that for all r ∈ (R1 , R2 ), Z 1 f (z) an = dz. 2πi C(z0 ,r) (z − z0 )n+1 9. Let fn be a sequence of holomorphic functions on U = {|z − z0 | < 3r}. Suppose that there exists a dense subset V of U on which fn converges. Suppose that for a positive number M , |fn (z)| ≤ M for all n and all z in {z : |z −z0 | ≤ 2r}. Prove that fn converges uniformly on {z : |z −z0 | ≤ r} to a holomorphic function. 9 Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 7 Example Sheet 7 1. Show P∞by an example the statement that a−n + · · · + an → s is not the same as n=−∞ an = s. 2. Deduce the Laurent series for f (z) = (a). D(0, 1); 1 (z−2)(z−3) in the following regions. (b). 2 < |z| < 3; (c). |z| > 3. 3. The following functions have singularities at 0. Describe the type of the singularity for each function. 3+z ; + 3z 2 ) z 4 (5 1 , tan z sin z ; z 1 cos( ); z ez z −1−z− z2 2 ; 4. If the order of the pole at z0 is n, prove that Res(f ; z0 ) = lim z→z0 1 d ( )n−1 (z − z0 )n f (z). (n − 1)! dz 5. Evaluate the residues of the following functions at their singularities: 1 , 1 + z2 6. Is sin(z) bounded on C? 1 , sin z ( z+1 2 ) z−1 1 ez . Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 8 10 Example Sheet 8. 1. (a) State Cauchy’s theorem for a closed curve γ ≈ 0. (b) Prove the following Cauchy’s theorem for R a simply connected domain: If U is a simply connected region then γ f (z)dz = 0 for every holomorphic function f : U → C and every closed piecewise C 1 curve γ in U . (c) What can you conclude for a star region? 2. How many zeros, counting multiplicity, does the function f (z) = z 7 −3z 3 + ez + 7 have in the annulus A = {z : 1 < |z| < 2}? 3. Hurwitz’s theorem. Let U be a region, fn : U → C a sequence of holomorphic functions converging to a function f uniformly on compact subsets of U . If each fn has no zero on U prove that f has no zero in U unless f is identically zero. rz e 4. Let f (z) = 1+e z where r ∈ (0, 1) is a real number and R a positive number. Let γ be the rectangular, oriented positively, with one side the segment [−R, R] on the real axis and a parallel side passing through 2πi. Find the poles of f inside the rectangular and compute their residues. 5. Prove that Z ∞ −∞ π erx dx = . x 1+e sin(πr) 6. Optional Question. Let γ be a piecewise smooth curve and f is defined and continuous on γ. For m ≥ 1 define for z 6∈ {γ}, Z f (ζ) gm (z) = dζ. m γ (ζ − z) 0 (z) = mgm+1 (z). Hint. Use Prove that g is holomorphic on C \ {γ} and gm the definition of complex differentiation. Complex Analysis (2016). Lecturer: Xue-Mei Li. Support Classes: Louis Bonthrone and Aljalila Al Abri 9 11 Example Sheet 9. Let D = {z : |z| = 1}. 1. (a) Prove Schwartz’s Lemma. If f : D → D is a holomorphic function with f (0) = 0, then either (1) f (z) = cz for a complex number c with |c| = 1 or (2) |f (z)| < |z| for all z ∈ {z : 0 < |z| < 1}. (b) Prove that every bi-holomorphic function f : D → D with f (0) = 0 is of the form f (z) = cz where |c| = 1. (c) Let b be such that |b| < 1. Prove that Gb : D → D given by the z−b formula Gb (z) = 1− is bi-holomorphic. b̄z (d) Prove every bi-holomorphic mapping from D onto D is of the form cGb for some c, b ∈ C such that |c| = 1 and |b| < 1. (e) Let U be a simply connected region which is not the whole plane and z0 ∈ U . Suppose that f, g : U → D are bi-holomorphic functions such that f (z0 ) = g(z0 ) = 0, f 0 (z0 ) > 0 and g 0 (z0 ) > 0. Prove that f = g. (Proposition 9.3.2) 2. Suppose that f is an entire function with no zero and for r > 0 define m(r) = inf |z|=r {|f (z)|}. Prove that m is non-increasing. 3. Prove that there is a homeomorphism (continuous map with continuous inz is a verse) from any simply connected region to C. Hint. g(z) = 1−|z| homeomorphism from D to C. 4. (a) Let f be an entire function that is not a polynomial. Given w ∈ C prove that for any > 0 and R > 0 there exists z with |z| > R such that |f (z) − w| < . Hint. Investigate the singularity of g(z) = f ( z1 ). (b) If f is a one-to one entire function, then f (z) = az + b for some complex numbers a, b ∈ C. 5. Suppose that f is a nowhere vanishing holomorphic function in a star region U . Prove that there exists a holomorphic function g s.t. eg(z) = f (z). 6. Let U be a region, fn : U → C a sequence of holomorphic functions converging to a non-constant function f uniformly on compact sets. Prove (1) Hurwitz’s Theorem. Let z0 ∈ U and w0 = f (z0 ). For any > 0 sufficiently small there is an integer N () > 0 such that for each n ≥ N (), there exists zn with |zn − z0 | < and fn (zn ) = w0 . (2) If each fn is one to one then so is f .