Applied and Computational Complex Analysis, Volume 3: Discrete Fourier Analysis, Cauchy Integrals, Construction of Conformal Maps, Univalent Functions
Presents applications as well as the basic theory of analytic functions of one or several complex variables. The first volume discusses applications and basic theory of conformal mapping and the solution of algebraic and transcendental equations. Volume Two covers topics broadly connected with ordinary differental equations: special functions, integral transforms, asymptotics and continued fractions. Volume Three details discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions, potential theory in the plane and polynomial expansions.
algorithm analytic function annulus applications approximation arbitrary assume boundary correspondence function boundary values bounded Cauchy integral closure compact subset complex compute condition conformal mapping conjugate harmonic function constant construct continuous function converges convolution Corollary curve F defined denote derivative differential Dirichlet problem discrete Fourier transform doubly connected region evaluated EXAMPLE exists Faber finite follows Fourier coefficients Fourier series function g given Green's function hence Hilbert transform Holder continuous holds integral equation interior of F inverse Jordan curve kernel Koebe Laurent series Lemma Let F linear logarithmic mapping function method modulus multiplications Neumann problem numerical obtain operator parameter piecewise analytic Poisson's equation principal value Privalov problem proof proved quadrilateral real function representation result satisfies sequence simply connected simply connected region singularity Sokhotskyi formulas solution solves Symm's uniformly unique unit disk yields zero