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    Stein and Shakarchi, Complex Analysis

    SRC-TEXT-SS03

    Area
    Complex Analysis
    Status
    draft
    Source
    Contents

    180 problems.

    Ch1 Exercises

    1. Geometric loci in the complex plane
    2. Euclidean and Hermitian inner products on \(\mathbb C\)
    3. The \(n\) solutions of \(z^n=\omega\)
    4. The complex numbers admit no ordered-field structure
    5. An open set is connected if and only if it is pathwise connected
    6. Connected components as equivalence classes
    7. Blaschke factors map the unit disc bijectively onto itself
    8. Chain rule for the Wirtinger derivatives \(\partial_z\) and \(\partial_{\bar z}\)
    9. The Cauchy-Riemann equations in polar coordinates
    10. Wirtinger operators factor the Laplacian
    11. Real and imaginary parts of holomorphic functions are harmonic
    12. Cauchy-Riemann equations at one point do not imply complex differentiability
    13. Holomorphic \(f\) with constant \(\Re f\), \(\Im f\), or \(\abs{f}\) is constant
    14. Summation by parts
    15. Abel’s theorem for convergent series
    16. Radii of convergence of explicit power, hypergeometric, and Bessel series
    17. The ratio test computes the radius of convergence
    18. A power series re-expands around every point of its disc of convergence
    19. Convergence of \(\sum kz^k\), \(\sum z^k/k^2\), and \(\sum z^k/k\) on the unit circle
    20. Power-series expansion and coefficient asymptotics of \((1-z)^{-m}\)
    21. Two dyadic series identities for \(z/(1-z)\)
    22. \(\NN\) is not a finite disjoint union of arithmetic progressions with distinct steps
    23. A smooth function whose Taylor series at \(0\) vanishes identically
    24. Reversing orientation negates a contour integral
    25. Integrals of \(z^n\) and \(1/((z-a)(z-b))\) over circles
    26. Primitives of a continuous function differ by a constant

    Ch2 Exercises

    1. The Fresnel integrals \(\int_0^\infty\sin(x^2)\,dx=\int_0^\infty\cos(x^2)\,dx=\frac{\sqrt{2\pi}}4\)
    2. \(\int_0^\infty\frac{\sin x}{x}\,dx=\frac\pi2\)
    3. Laplace transforms of \(\cos bx\) and \(\sin bx\) by a sector contour
    4. The Gaussian \(e^{-\pi x^2}\) is its own Fourier transform
    5. Goursat’s theorem for \(C^1\) functions via Green’s theorem
    6. Cauchy’s theorem for a triangle with one bounded interior singularity
    7. \(2\abs{f'(0)}\) is at most the diameter of \(f(\mathbb D)\)
    8. Derivatives inherit polynomial growth on a strip
    9. A self-map of a bounded set with \(\varphi(z_0)=z_0\) and \(\varphi'(z_0)=1\) is the identity
    10. Not every continuous function on the closed disc is a uniform limit of polynomials in \(z\)
    11. The Cauchy formula with kernel \(\Re\frac{\zeta+z}{\zeta-z}\) and the Poisson kernel
    12. Harmonic conjugates and Poisson representation on the unit disk
    13. An entire function with a vanishing Taylor coefficient at every point is a polynomial
    14. A pole on the circle of convergence and its coefficient growth
    15. A nonvanishing function holomorphic on the disc with \(\abs f=1\) on the circle is constant

    Ch2 Problems

    1. Natural boundaries of the lacunary series \(\sum z^{2^n}\) and \(\sum 2^{-n\alpha}z^{2^n}\)
    2. Growth of \(\sum d(n)z^n\) along rational rays
    3. Morera’s theorem for circles and toy contours

    Ch3 Exercises

    1. Zeros of \(\sin\pi z\) and residues of \(1/\sin\pi z\)
    2. \(\int_{-\infty}^\infty\frac{dx}{1+x^4}=\frac{\pi}{\sqrt2}\)
    3. \(\int_{-\infty}^\infty\frac{\cos x}{x^2+a^2}\,dx=\pi\frac{e^{-a}}{a}\)
    4. \(\int_{-\infty}^\infty\frac{x\sin x}{x^2+a^2}\,dx=\pi e^{-a}\)
    5. The Fourier transform of \((1+x^2)^{-2}\)
    6. \(\int_{-\infty}^\infty\frac{dx}{(1+x^2)^{n+1}}=\frac{1\cdot3\cdots(2n-1)}{2\cdot4\cdots(2n)}\pi\)
    7. \(\int_0^{2\pi}\frac{d\theta}{(a+\cos\theta)^2}=\frac{2\pi a}{(a^2-1)^{3/2}}\)
    8. \(\int_0^{2\pi}\frac{d\theta}{a+b\cos\theta}=\frac{2\pi}{\sqrt{a^2-b^2}}\)
    9. \(\int_0^1\log\sin\pi x\,dx=-\log2\)
    10. \(\int_0^\infty\frac{\log x}{x^2+a^2}\,dx=\frac{\pi}{2a}\log a\)
    11. \(\int_0^{2\pi}\log\abs{1-ae^{i\theta}}\,d\theta=0\) for \(\abs a\le1\)
    12. \(\sum_{n\in\ZZ}(u+n)^{-2}=\pi^2/\sin^2\pi u\) by residues
    13. A mild pole bound forces a removable singularity
    14. Injective entire functions are affine
    15. Applications of the Cauchy inequalities and the maximum modulus principle
    16. The zero of \(f+\epsilon g\) near a simple zero of \(f\) is unique and continuous in \(\epsilon\)
    17. Image of a non-constant holomorphic map containing the unit disc
    18. The Cauchy integral formula via homotopy
    19. The maximum principle for harmonic functions
    20. \(L^2\) norms control sup norms of holomorphic functions on smaller discs
    21. Convex and star-shaped open sets are simply connected
    22. No holomorphic function on the disk can extend continuously with boundary values \(1/z\)

    Ch3 Problems

    1. Koebe-Bieberbach radius theorem for normalized univalent functions
    2. Poisson’s integral formula for the disk

    Ch4 Exercises

    1. A continuous function of moderate decrease with \(\hat f=0\) vanishes
    2. Derivatives of functions in \(\mathfrak F_a\) lie in \(\mathfrak F_b\) for \(b<a\)
    3. The Fourier transform pair \(\frac1\pi\frac{a}{a^2+x^2}\) and \(e^{-2\pi a\abs\xi}\)
    4. The Fourier transform of \(1/Q(x)\) in terms of the roots of \(Q\)
    5. Fourier transforms of rational functions without real poles
    6. \(\frac1\pi\sum_n\frac{a}{a^2+n^2}=\coth\pi a\) by Poisson summation
    7. \(\sum_n(\tau+n)^{-k}\) and \(\sum_n(\tau+n)^{-2}=\pi^2/\sin^2\pi\tau\) by Poisson summation
    8. Taylor coefficients of a function whose Fourier transform has compact support
    9. Phragmén–Lindelöf theorems in a half-plane and a sector
    10. Gaussian decay in a strip passes to the Fourier transform
    11. Entire functions of order 2 with Gaussian decay on the real line
    12. Hardy’s theorem: \(f\) and \(\hat f\) both \(O(e^{-\pi x^2})\) force \(f=ce^{-\pi x^2}\)

    Ch4 Problems

    1. Exponential decay of the Fourier transform gives holomorphy
    2. SS 4.PR-2: Solving a linear ODE with the Fourier transform

    Ch5 Exercises

    1. Jensen’s formula via Blaschke factors
    2. Orders of growth of \(p(z)\), \(e^{bz^n}\) and \(e^{e^z}\)
    3. The Jacobi theta function has order \(2\)
    4. Growth and zeros of \(\prod_n(1-e^{-2\pi nt}e^{2\pi iz})\)
    5. \(\int_\RR e^{-\abs t^\alpha}e^{2\pi izt}\,dt\) has order \(\alpha/(\alpha-1)\)
    6. Wallis’s product formula
    7. Convergence of \(\prod(1+a_n)\) compared with \(\sum a_n\)
    8. \(\prod_{k\ge1}\cos(z/2^k)=\sin z/z\)
    9. \(\prod_{k\ge0}(1+z^{2^k})=\frac1{1-z}\) for \(\abs z<1\)
    10. Hadamard products of \(e^z-1\) and \(\cos\pi z\)
    11. Picard’s little theorem for entire functions of finite order
    12. A zero-free entire function of finite order with zero-free derivatives is \(e^{az+b}\)
    13. \(e^z=z\) has infinitely many solutions
    14. An entire function of non-integral order has infinitely many zeros

    Ch5 Problems

    1. Zeros of a bounded holomorphic function satisfy the Blaschke condition
    2. Infinite Blaschke products from sequences satisfying the Blaschke condition
    3. Order of an entire function with coefficients \((n!)^{-\alpha}\)
    4. Growth order of an entire function from its Taylor coefficients

    Ch6 Exercises

    1. Gauss’s limit formula for \(\Gamma(s)\)
    2. \(\prod_n\frac{n(n+a+b)}{(n+a)(n+b)}=\frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+1)}\)
    3. Wallis’s product and the duplication formula for \(\Gamma\)
    4. Taylor coefficients of \((1-z)^{-\alpha}\) via \(\Gamma\)
    5. \(\abs{\Gamma(1/2+it)}^2=\frac{2\pi}{e^{\pi t}+e^{-\pi t}}\)
    6. \(1+\frac13+\cdots+\frac1{2n-1}-\frac12\log n\to\frac\gamma2+\log2\)
    7. The Beta function and \(B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\)
    8. The power series of the Bessel function \(J_\nu\)
    9. An integral representation of the hypergeometric series
    10. Mellin transforms of \(\cos t\) and \(\sin t\)
    11. The Fourier transform of \(e^{az}e^{-e^z}\) is \(\Gamma(a-2\pi i\xi)\)
    12. \(1/\Gamma\) is not of exponential type
    13. \(\frac{d^2}{ds^2}\log\Gamma(s)=\sum_{n\ge0}(s+n)^{-2}\)
    14. An asymptotic formula for \(\log n!\)
    15. \(\zeta(s)=\frac1{\Gamma(s)}\int_0^\infty\frac{x^{s-1}}{e^x-1}\,dx\)
    16. Analytic continuation of \(\zeta\) from its Gamma-integral representation
    17. Analytic continuation of the Mellin transform of a Schwartz function

    Ch6 Problems

    1. Estimates for \(\zeta\) and \(\zeta'\) near \(\Re(s)=1\)
    2. \(\zeta(s)=\frac{s}{s-1}-s\int_1^\infty\{x\}x^{-s-1}\,dx\) for \(\Re(s)>0\)
    3. Analytic continuation of \(\zeta\) to \(\Re(s)>-k\) by periodic antiderivatives of \(\{x\}-\frac12\)
    4. The periodic antiderivatives \(Q_k\) are Bernoulli polynomials, and \(\zeta(2k)\) in terms of \(B_{2k}\)

    Ch7 Exercises

    1. Bounded partial sums give convergence of the Dirichlet series for Re(s) > 0
    2. Products of Dirichlet series and Dirichlet convolution
    3. \(\sum_n\mu(n)n^{-s}=1/\zeta(s)\) for the Möbius function
    4. Dirichlet L-series for periodic coefficients

    Ch7 Problems

    1. Mean square of a Dirichlet series with bounded coefficients, and uniqueness of coefficients
    2. The explicit formula for \(\psi_1(x)\) as a sum over zeros of \(\zeta\)
    3. \(\pi(x)-\operatorname{Li}(x)=O(x^{\alpha+\epsilon})\) and the zero-free half-plane \(\Re s>\alpha\)
    4. The prime number theorem in arithmetic progressions

    Ch8 Exercises

    1. A holomorphic map is a local bijection exactly where \(f'\ne0\)
    2. Orthogonal curves through a nondegenerate critical point of \(F\)
    3. Simple connectedness is a conformal invariant
    4. No holomorphic surjection from the disc onto \(\CC\)
    5. A conformal map from the half-disc onto the upper half-plane
    6. Harmonicity is preserved by conformal maps
    7. The solution of the Dirichlet problem in a strip
    8. A harmonic function on the first quadrant with piecewise constant boundary values
    9. An unbounded harmonic function with zero boundary values
    10. A Schwarz-type bound on the upper half-plane
    11. \(\abs{\frac{f(z)-f(0)}{M^2-\overline{f(0)}f(z)}}\le\frac{\abs z}{MR}\) for \(\abs f<M\) on \(D(0,R)\)
    12. Fixed points of holomorphic self-maps of the disc
    13. The pseudo-hyperbolic metric and the Schwarz–Pick inequality
    14. Conformal maps from the upper half-plane onto the disc
    15. An automorphism of \(\HH\) is determined by the images of three boundary points
    16. Automorphisms of the disc through the Cayley transform
    17. The area integrals of \(\abs{\psi_\alpha'}^2\) and \(\abs{\psi_\alpha'}\) over the disc
    18. Continuous extension of a conformal map of the disc to the boundary
    19. The plane slit along parallel rays is simply connected
    20. Elliptic integrals mapping the upper half-plane onto rectangles
    21. Schwarz–Christoffel maps onto triangles
    22. The Schwarz–Christoffel formula for maps from the disc onto a polygon
    23. A conformal map onto a regular polygon
    24. Identities between the elliptic integrals \(K\) and \(K'\)

    Ch8 Problems

    1. Isogonality and isotropy at a point
    2. Holomorphic maps with \(f'(z_0)\ne0\) preserve angles, and angle-preserving maps are holomorphic
    3. Hyperbolic metric on the disc and the Schwarz-Pick lemma

    Ch9 Exercises

    1. A meromorphic function with two periods of real ratio
    2. The zeros minus the poles of an elliptic function lie in the period lattice
    3. \(\sum_{\omega\in\Lambda^*}\abs\omega^{-2}\) diverges
    4. Periodicity of \(\wp\) by rearranging its series
    5. The Weierstrass sigma function
    6. The differential equation of \(\wp\)
    7. \(\zeta(2)\) and \(\zeta(4)\) from \(\sum_m(m+\tau)^{-2}=\pi^2/\sin^2\pi\tau\)
    8. \(E_4(\tau)\to\pi^4/45\) as \(\Im\tau\to\infty\), with exponential error

    Ch9 Problems

    1. Besides the approach in Section 1

    Ch10 Exercises

    1. A theta-quotient identity for \(\wp_\tau\)
    2. The generating function of the Fibonacci numbers
    3. Solving \(u_n=au_{n-1}+bu_{n-2}\) by generating functions
    4. Euler’s pentagonal recurrence for the partition function
    5. Logarithmic asymptotics of the partition generating function
    6. Bounds \(e^{c_1\sqrt n}\le p(n)\le e^{c_2\sqrt n}\) for the partition function
    7. Triangular and septagonal number identities from the Jacobi triple product
    8. Parametrization of primitive Pythagorean triples
    9. Sums of two squares from the formula for \(r_2(n)\)
    10. Irregular growth of \(r_2(n)\) and \(r_4(n)\)
    11. Lambert series for the divisor-power sums \(\sigma_\ell(n)\)
    12. A Lambert-series formula for \(\theta^4\) equivalent to the four-squares theorem

    Ch10 Problems

    1. \(S\) and \(T_1\) generate the modular group
    2. \(S\) and \(T_2\) generate the theta group
    3. Bounded holomorphic functions invariant under the theta group are constant

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