Complex Analysis Qual Sheet Robert Won \Tricks and traps. Basically all complex analysis qualifying exams are collections of tricks and traps." - Jim Agler 1 Useful facts 1 X zn 1. ez = n! n=0 1 X z2n+1 1 2. sin z = (−1)n = (eiz − e−iz) (2n + 1)! 2i n=0 1 X z2n 1 3. cos z = (−1)n = (eiz + e−iz) 2n! 2 n=0 1 4. If g is a branch of f −1 on G, then for a 2 G, g0(a) = f 0(g(a)) 5. jz ± aj2 = jzj2 ± 2Reaz + jaj2 6. If f has a pole of order m at z = a and g(z) = (z − a)mf(z), then 1 Res(f; a) = g(m−1)(a): (m − 1)! 7. The elementary factors are defined as z2 zp E (z) = (1 − z) exp z + + ··· + : p 2 p Note that elementary factors are entire and Ep(z=a) has a simple zero at z = a. 8. The factorization of sin is given by 1 Y z2 sin πz = πz 1 − : n2 n=1 9. If f(z) = (z − a)mg(z) where g(a) 6= 0, then f 0(z) m g0(z) = + : f(z) z − a g(z) 1 2 Tricks 1. If f(z) nonzero, try dividing by f(z). Otherwise, if the region is simply connected, try writing f(z) = eg(z). 2. Remember that jezj = eRez and argez = Imz. If you see a Rez anywhere, try manipulating to get ez. 3. On a similar note, for a branch of the log, log reiθ = log jrj + iθ. 4. Let z = eiθ. 5. To show something is analytic use Morera or find a primitive. 6. If f and g agree on a set that contains a limit point, subtract them to show they're equal. 7. Tait: \Expand by power series." 8. If you want to count zeros, either Argument Principle or Rouch´e. 9. Know these M¨obiustransformations: 1 − z (a) To map the right half-plane to the unit disk (or back), . 1 + z z − a (b) To map from the unit disk to the unit disk, remember ' (z) = . This is a bijective a 1 − az 1 − jaj2 map with inverse ' (z). Also, ' (a) = 0, '0 (z) = , '0 (0) = 1 − jaj2, and −a a a (1 − az)2 a 1 '0 (a) = . a 1 − jaj2 10. If f(z) is analytic, then f(z) is analytic (by Cauchy-Riemann). So if, for example, f(z) is real on the real axis, then f(z) = f(z). 11. To prove that a function defined by an integral is analytic, try Morera and reversing the R 1 −t z−1 R R 1 −t z−1 R 1 R −t z−1 integral. (e.g. e t dt is analytic since T e t dtdz = T e t dzdt = 0:) 12. If given a point of f (say f(0) = a) and some condition on f 0 on a simply connected set, try R 0 [0;z] f = f(z) − f(0). 13. To create a non-vanishing function, consider exponentiating. 3 Theorems 1. Cauchy Integral Formula: Let G be region and f : G ! C be analytic. If γ1; : : : ; γm Pm are closed rectifiable curves in G with k=0 n(γk; w) = 0 for all w 2 C n G, then for a 2 m G n ([k=1fγkg), m m X n! X Z f(z) f (n)(a) · n(γ ; a) = dz: k 2πi (z − a)n+1 k=0 k=1 γk 2 2. Cauchy's Theorem: Let G be a region and f : G ! C be analytic. If γ1; : : : ; γm are closed Pm rectifiable curves in G with k=0 n(γk; w) = 0 for all w 2 C n G, then m X Z f(z)dz = 0 k=1 γk 3. Liouville's Theorem: If f is a bounded entire function, then f is constant. 4. Maximum Modulus Theorem: Let G be a region and f : G ! C be analytic. If there exists an a 2 G such that jf(a)j ≥ jf(z)j for all z 2 G, then f is constant on G. R 5. Morera's Theorem: Let G be a region and f : G ! C be continuous. If T f = 0 for every triangular path T in G, then f is analytic on G. 6. Goursat's Lemma: Let G be a region and let f : G ! C. If f is differentiable, then f is analytic on G. 7. Cauchy-Riemann Equations: Let f(x; y) = u(x; y) + iv(x; y) for real-valued functions u and v. Then f is analytic if and only if @u @v @u @v = and = − @x @y @y @x 8. Constant functions: Let f : G ! C be analytic. Then the following are equivalent (i) f(z) ≡ α; (ii) fz 2 G j f(z) = αg has a limit point in G; (iii) there exists a 2 G such that f (n)(a) = 0 for all n ≥ 1. 0 9. Conformality: Let f : G ! C be analytic. Then if z 2 G and f (z) 6= 0, f is conformal at z. 10. Roots of an analytic function: Let f : G ! C be analytic. If f(a) = 0, then there exists a unique m ≥ 1 and g analytic such that f(z) = (z − a)mg(z) with g(a) 6= 0. 11. Power series: A function f is analytic on B(a; R) if and only if there exists a power series P1 n f(z) = n=0 an(z − a) where we compute f (n)(a) 1 Z f(z) an = = n+1 dz: n! 2πi γ (z − a) The series converges absolutely on B(a; R) and uniformly on B(a; r) for 0 ≤ r < R. 12. Cauchy's Estimate: If f analytic on B(a; R), and jf(z)j ≤ M for each z 2 B(a; R), then n!M f (n)(a) ≤ : Rn 3 13. Winding Number: To compute the index of a closed curve about a point a, 1 Z dz n(γ; a) = 2 Z: 2πi γ z − a 14. Open Mapping Theorem: Let G be a region, f a non-constant analytic function. If U is an open subset of G, then f(U) is open. 15. Zero-Counting Theorem: Let G be a region, f : G ! C analytic with roots a1; : : : am. If fγg ⊆ G and ak 62 fγg for all k, and γ ≈ 0 in G, then m 1 Z f 0(z) X dz = n(γ; a ) 2πi f(z) k γ k=1 Corollary: If f(a) = α, then f(z) − α has a root at a. So if f(ak) = α, then m 1 Z f 0(z) X dz = n(γ; a ) 2πi f(z) − α k γ k=1 Corollary 2: If σ = f ◦ γ and α 62 fσg and ak are the points where f(ak) = α, then m X n(σ; α) = n(γ; ak) or k=1 m X n(f ◦ γ; f(a)) = n(γ; ak) k=1 16. Roots of analytic functions: Suppose f is analytic on B(a; R) and let f(a) = α. If f(z)−α has a zero of order m at z = a, then there exist > 0 and δ > 0 such that if 0 < jζ − αj < δ, the equation f(z) = ζ has exactly m simple roots in B(a; ). 17. Existence of Logarithm: Let f(z) be analytic and f(z) 6= 0 on G, a simply connected region. Then there is analytic function g(z) on G such that f(z) = eg(z) for all z 2 G. 18. Existence of Primitive: Let f(z) be analytic on G, a simply connected region. Then f has a primitive. 19. Laurent Series: Let f be analytic on R1 < jz − aj < R2, then there exists a sequence 1 fangn=−∞ and 1 X n f(z) = an(z − a) n=−∞ with absolute convergence in the open annulus and uniform convergence on every compact subset of the annulus. This series is called a Laurent series, and if γ is a closed curve in the annulus, then 1 Z f(z) an = n+1 dw: 2πi γ (z − a) (Note that this is just the same as number 11). 4 20. Classification of Singularities: Suppose f analytic on B(a; R) n fag and f has an isolated singularity at a. Then a is (a) Removable singularity if there is a function g analytic on B(a; R) such that f(z) = g(z) for all z 2 B(a; R) n fag. The singularity is removable if and only if lim(z − a)f(z) = 0. z!a Also, the singularity is removable if and only if the Laurent series of f has no coefficients an for n < 0. (b) Pole if lim jf(z)j = 1. z!a If a is a pole, then there is a unique m ≥ 1 and an analytic function g such that g(z) f(z) = for all z 2 B(a; R) n fag and g(a) 6= 0. (z − a)m The singularity is a pole if and only if the Laurent series of f has only finitely many coefficients an for n < 0. The partial series for these coefficients is called the singular part of f. (c) Essential singularity if a is not removable and not a pole. The singularity is essential if and only if the Laurent series of f has infinitely many coefficients an for n < 0.