Search the corpus

    McMullen, Harvard Math 213a Advanced Complex Analysis exercises (2017)

    SRC-HARVARD-MATH213A-2017

    Area
    Complex Analysis
    Status
    draft
    Source
    Contents

    156 problems.

    Review exercises

    1. Stokes theorem on a spherical triangle
    2. Fifth compositional roots of a Möbius transformation
    3. Taylor coefficients and numerical approximation of tanh
    4. A proper local homeomorphism is a covering map
    5. Polynomial maps away from critical values are coverings
    6. A polynomial giving a normal covering off its critical values

    Embedded exercises

    1. The figure-eight knot is not an analytic plane-curve link
    2. Spherical distance from 1 to 1+i
    3. A continuous image of the circle is locally connected

    Section 1.8 exercises

    1. Complex multiplication from dot and planar cross products
    2. Area enclosed by an analytic image of the unit circle
    3. Wirtinger derivatives of a polar power-angle map
    4. Mixed Wirtinger derivative of the squared modulus
    5. Characterize polynomials splitting into holomorphic and antiholomorphic parts
    6. Analytic maps into strictly convex level sets are constant
    7. Zeros persist under compact-uniform convergence
    8. Distributional Wirtinger derivative of one over z
    9. Evaluate a rational integral by complex methods
    10. Constant boundary modulus forces a zero unless the function is constant
    11. Laurent expansion on the annulus between one and two
    12. A polynomial cannot uniformly approximate one over z on the unit circle
    13. Nonzero boundary degree forces an interior zero
    14. Generating function for partitions into unequal parts
    15. Sum of reciprocal derivatives at simple polynomial roots
    16. Removable compact null sets on a diameter
    17. Pointwise limits of polynomials are analytic on a dense open set
    18. Compare two power-series approximations to log two
    19. Essential singularities with prescribed logarithmic residue
    20. A beta-type improper integral and its convergence range
    21. Vanishing of the upper semicircle exponential integral
    22. Residue derivation of the even cosine-power integral
    23. Prescribe a cluster value of the exponential essential singularity
    24. First four nonzero terms of the tangent series by formal inversion
    25. Analytic functions with equal modulus differ by a unimodular constant
    26. Homogeneous harmonic polynomials on the plane
    27. Equal L2 norms of real and imaginary parts of a disk analytic function
    28. Pointwise bound from the L1 norm of an analytic function
    29. Uniform limits of harmonic functions are harmonic
    30. Bounded harmonic function with opposite semicircle boundary values
    31. A bounded harmonic function with unbounded harmonic conjugate
    32. Strictly subharmonic functions satisfy the maximum principle
    33. Unbounded analytic function with unit boundary modulus except at one point

    Section 2.5 exercises

    1. There is no proper holomorphic map from the disk to the plane
    2. Proper holomorphic maps involving the punctured plane
    3. Quotient maps for finite Möbius groups
    4. Spherical metric invariance and generators of PSU two
    5. Reduction and radical solution of a generic cubic
    6. Zeros critical points and critical values of Chebyshev polynomials
    7. Normal form for proper degree-two disk maps
    8. Conformal map from a half-infinite rectangle to the upper half-plane
    9. Stereographic coordinates and great-circle distance
    10. Partial-fraction form of proper upper-half-plane self-maps
    11. Iterates of a disk self-map fixing the origin
    12. Hyperbolic distance in projective coordinates
    13. Area of a hyperbolic triangle from its angles
    14. Hyperbolic metric on the punctured disk
    15. Covering map and hyperbolic geometry of a round annulus
    16. Boundary behavior of iterates without an interior fixed point
    17. Growth bound for analytic functions with nonnegative real part
    18. Proper disk maps preserve boundary averages
    19. Compactness of normalized positive harmonic functions on the disk
    20. No conformal metric on the sphere is invariant under every Möbius transformation
    21. The Lie algebra of \(SL_2(\mathbb C)\) as holomorphic vector fields on \(\mathbb P^1\)
    22. Subsequence degeneration of unbounded Möbius transformations
    23. Degree-three rational maps preserving the unit circle and converging to the identity
    24. Circular triangles with equal ordered angles are Möbius equivalent

    Section 3.4 exercises

    1. Polynomial minus exponential has infinitely many zeros — Section 3.4 Exercise 1
    2. Equality in three-circles convexity forces a monomial
    3. A nontrivial bounded holomorphic disk function cannot vanish at \(1-1/(n+1)\)
    4. Infinite product for \(\sin^2(\sqrt z)\)
    5. A genus-zero canonical product of exact order one
    6. A theta series is entire of order two
    7. Interpolation by entire functions at prescribed points — Section 3.4 Exercise 7
    8. Bézout identity for entire functions with no common zeros
    9. Entire functions are integrally closed among meromorphic functions
    10. Entire functions of bounded order form an integrally closed ring
    11. Every finitely generated ideal of the entire-function ring is principal
    12. A nonprincipal ideal in the ring of entire functions
    13. A fourth-order partial-fraction sum and \(\zeta(4)\)
    14. A nonzero entire function bounded by a prescribed positive function on the real axis
    15. A nontrivial bounded holomorphic disk function cannot vanish at \(1-1/(n+1)\) — Section 3.4 Exercise 15 (repeats Exercise 3)
    16. Evaluate a hyperbolic-cotangent Dirichlet series by residues
    17. Orders of \(\sin^2(\sqrt z)\) and \(e^{\sin z}\)
    18. Quasi-periodicity of an entire function with simple square-lattice zeros
    19. Evaluate \(\int_0^\infty e^{-x^\alpha}\,dx\) using the Gamma function
    20. An asymptotic formula for \(\Gamma'(x)\) on the positive real axis
    21. Decide whether \(\Gamma'(3/2)=0\)
    22. \(1/\Gamma\) has order one but no exponential-type bound
    23. The Gamma triplication formula
    24. Evaluate \(\int_0^1\log\Gamma(t)\,dt\)
    25. Solve \(f(z+1)=p(z)f(z)\) meromorphically for a polynomial \(p\)
    26. Characterize logarithmic derivatives of meromorphic functions on \(\mathbb C\)
    27. Construct an entire function of order \(1/3\)
    28. Build an entire function with prescribed zeros from generalized convergence factors
    29. Wallis’s product formula — Section 3.4 Exercise 29
    30. If nonconstant entire functions satisfy \(f\circ g=f\), then \(g\) is affine
    31. Finite order without convergence of the logarithmic growth quotient
    32. Estimate the average of \(\sin^{100}x\) by stationary phase
    33. Entire functions above and below a prescribed positive real-axis profile
    34. Shift the harmonic sum to obtain \(\log n+o(1)\)

    Section 4.4 exercises

    1. Normalized univalent maps cannot have strictly nested images
    2. A second-coefficient bound for bounded normalized univalent maps
    3. Omitted values of a normalized exterior univalent map lie in the disk of radius two
    4. Uniformization of plane regions with at least three complementary points
    5. Compactness of normalized univalent maps on a fixed plane region
    6. Conformally map a round annulus to the region between an ellipse and its focal segment
    7. Length-area inequality for transverse curves in a conformal annulus
    8. Finite Dirichlet integral implies almost-everywhere finite radial limits
    9. A hyperbolic distance-decreasing disk homeomorphism with no continuous boundary extension
    10. Conformal radii of standard pointed regions
    11. Map the exterior disk conformally onto the complement of a semicircle
    12. Jordan-domain Riemann maps are dense among normalized univalent maps
    13. Map the exterior disk conformally onto the exterior of an ellipse
    14. Uniqueness questions for simply connected regions of prescribed conformal radius
    15. Can a normalized exterior univalent map have \(b_2=1/\sqrt2\)?
    16. Polygonal Riemann maps are dense among normalized univalent maps
    17. Kernel convergence can shrink the image of the limiting Riemann map
    18. Fourfold symmetry improves the guaranteed disk in a normalized univalent image
    19. Elementary coefficient bounds for normalized univalent functions
    20. Derive the Beta-Gamma identity from the Gamma functional equation
    21. Largest unramified disk for \(e^{z^2}\) on the unit disk
    22. Chain rule for the nonlinearity of a holomorphic map
    23. Chain rule for the Schwarzian derivative
    24. Wronskians and Schwarzians of ratios of solutions to \(u''+\phi u=0\)
    25. Cover the complement of a square by the upper half-plane
    26. Little Picard gives a uniform derivative bound into the twice-punctured plane
    27. Plane tilings and accumulation for groups generated by triangle reflections
    28. Entire solutions of \(f^n+g^n=1\) are constant for \(n>2\)
    29. A zero-free entire function taking the value one only finitely often is constant
    30. Quadratic differential with prescribed residues at \(0\), \(1\), and \(\infty\)

    Section 5.5 exercises

    1. Discrete subgroups of \(\mathbb R^n\) are free abelian and have compact quotient exactly in full rank
    2. Do successive shortest independent lattice vectors form a basis?
    3. Complex multiplication endomorphisms of a complex torus and their degrees
    4. Zeros of the Weierstrass \(\wp\)-function for square and hexagonal lattices
    5. Vanishing Eisenstein sums for square and hexagonal lattices
    6. Nonconstant meromorphic solutions of \(f^3+g^3=1\)
    7. Double-angle formula for the Weierstrass \(\wp\)-function
    8. Dense orbits for torus doubling and its Weierstrass quotient
    9. Do the weight-four and weight-six lattice sums determine a lattice?
    10. Lattice shape from a real Weierstrass cubic with one real root
    11. Map a right triangle to the upper half-plane using a Weierstrass \(\wp\)-function
    12. Weierstrass zeta as the logarithmic derivative of sigma
    13. Quasi-periods of the Weierstrass zeta function and the Legendre relation
    14. Quasi-periodicity of the Weierstrass sigma function
    15. Mittag-Leffler criterion for prescribed principal parts on a complex torus
    16. Express \(\wp'\) as a quotient of Weierstrass sigma functions
    17. Derive the modular function from the cross-ratio of the Weierstrass roots
    18. Automorphism groups of four-punctured spheres
    19. Hyperbolic metric near a puncture of the thrice-punctured sphere
    20. Length of the figure-eight geodesic on the thrice-punctured sphere
    21. Determine the universal relation between \(G_2^2\) and \(G_4\)
    22. Cusp orbits of \(\mathrm{SL}_2(\mathbb Z)\) and \(\Gamma(2)\)
    23. Rational modular forms for \(\Gamma(2)\) as a graded ring
    24. Evaluate the modular lambda function at \(i/2\)
    25. Evaluate the weight-four square-lattice Eisenstein sum using Gamma values
    26. Relate the square-lattice Eisenstein value to the conformal radius of the square

    Browse these problems in source order