harmonic function theory.zip

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  • 2020-06-14 22:38
谐波函数理论 mathematica .Axler
harmonic function theory.zip
  • HFT.nb
  • HFT.m
  • ComputingWithHarmonicFunctions.nb
(******************************************************************* This file was generated automatically by the Mathematica front end. It contains Initialization cells from a Notebook file, which typically will have the same name as this file except ending in ".nb" instead of ".m". This file is intended to be loaded into the Mathematica kernel using the package loading commands Get or Needs. Doing so is equivalent to using the Evaluate Initialization Cells menu command in the front end. DO NOT EDIT THIS FILE. This entire file is regenerated automatically each time the parent Notebook file is saved in the Mathematica front end. Any changes you make to this file will be overwritten. ***********************************************************************) If[$VersionNumber<4, Print["This package will not work correctly with versions of Mathematica \ earlier than Mathematica 4.0. If you need a version of this package that will \ work correctly with earlier versions of Mathematica, please contact Sheldon \ Axler at axler@sfsu.edu."]] Print["HFT; Version 6.10, 9 August 2003"] Print["This Mathematica package, called HFT.m, is designed for computing with \ harmonic functions. Documentation for the use of this package and information \ about the algorithms used in it is available in the Mathematica notebook \ called HFT.nb."] Print["For addional information about harmonic functions, see the book \ Harmonic Function Theory, by Sheldon Axler, Paul Bourdon, and Wade Ramey, \ published by Springer."] Print["This package is copyrighted by Sheldon Axler but is distributed \ without charge. The most recent version of this HFT.m package along with its \ documentation notebook HFT.nb is available over the world wide web at: \ http://www.axler.net"] Print["Comments, suggestions, and bug reports should be sent by electronic \ mail to: axler@sfsu.edu"] Print["* The computer is now unpacking the HFT.m package."] BeginPackage["HFT`"] Annulus::"usage"="Annulus is an option for Region in Dirichlet." AntiLaplacian::"usage"= "AntiLaplacian[f, x] gives a polynomial "<>"whose Laplacian equals f. "<> "Here f must be a polynomial function of x." Ball::"usage"= "Ball is one of the values that may be assigned to the option "<> "Orthonormal." BasisH::"usage"= "BasisH[m, x] gives a basis for the space of harmonic "<> "polynomials homogeneous of degree m in the variable x." BergmanKernel::"usage"= "BergmanKernel[x, y] gives the Bergman "<> "reproducing kernel for the unit ball." BergmanKernelH::"usage"= "BergmanKernel[z, w] gives the Bergman "<> "reproducing kernel for the upper half-space." BergmanProjection::"usage"= "BergmanProjection[ f, x] gives the "<> "orthogonal projection of a polynomial f, as a function of x, "<> "onto the Bergman space of harmonic functions on the ball." BiDirichlet::"usage"= "BiDirichlet[f , x] solves the "<> "BiDirichlet problem with boundary data f, "<>"as a function of x." Delta::"usage"= "Delta[j] is the vector that equals 1 in the j-th "<> "coordinate and 0 in the other coordinates." Dimension::"usage"="Dimension[x] is the dimension of the vector x." DimensionH::"usage"= "DimensionH[m, n] gives the vector space dimension of the "<> "space of spherical harmonics of degree m in n variables." Dirichlet::"usage"= "Dirichlet[f, x] gives the harmonic function "<> "that equals f on the unit sphere. Here f must be a "<> "polynomial function of x." Divergence::"usage"= "Divergence[f, x] gives the divergence of f "<>"with respect to x." DoubleBracketingBar::"usage"="DoubleBracketingBar is equal to the Norm." ExpandNorm::"usage"= "ExpandNorm[f] gives f with all terms of the form |x + y| "<> "replaced by Sqrt[ |x|^2 + |y|^2 + 2 x.y ]." ExteriorSphere::"usage"= "ExteriorSphere is an option for Dirichlet, "<> "specifying that the region should be the exterior " <> "of the unit sphere." ExteriorNeumann::"usage"= "ExteriorNeumann[f , x] solves the "<> "exterior Neumann problem with boundary data f, "<>"as a function of x." Grad::"usage"="Grad[f, x] gives the gradient of f with "<>"respect to x." HarmonicConjugate::"usage"= "HarmonicConjugate[u, x, y] gives "<> "the harmonic conjugate of u with respect to the variables x, y." HarmonicDecomposition::"usage"= "HarmonicDecomposition[u, x] "<> "gives the harmonic decomposition of u with respect to "<> "the variable x." HilbertSchmidt::"usage"= "HilbertSchmidt[A] gives the "<>"HilbertSchmidt norm of a matrix A." Homogeneous::"usage"= "Homogeneous[f, d, x] gives the homogenous "<> "part of f of degree d, with respect to x." Hyperplane::"usage"="Hyperplane[b, c] denotes the hyperplane "<>"b.x = c." IntegrateBall::"usage"= "IntegrateBall[f, x] gives the integral "<> "of f, as a function of x, over the unit ball." IntegrateSphere::"usage"= "IntegrateSphere[f, x] gives the "<> "integral of f, as a function of x, over the unit sphere "<> "with respect to normalized surface area measure." Reflection::"usage"="Reflection[x] gives the reflection of x "<> "in the unit sphere." J::"usage"="J[f, x] gives the Jacobian of f with "<>"respect to x." Kelvin::"usage"= "Kelvin[u, x] gives the Kelvin transform of u, "<> "thought of as a function of x." KelvinM::"usage"= "KelvinM is the modified Kelvin transform, "<> "as defined in Chapter 7 of Harmonic Function Theory." Laplacian::"usage"= "Laplacian[f, x] gives the Laplacian of f "<>"with respect to x." Multiple::"usage"= "Multiple is an option for AntiLaplacian. "<> "The default value is None. The value Norm^2 produces the "<> "unique antiLaplacian that is a polynomial multiple of "<> "Norm[x]^2, where x is the variable." Neumann::"usage"= "Neumann[f, g, x] solves the "<> "Neumann problem of finding a function of x whose "<> "outward normal derivative (on the unit sphere) is f "<> "and whose Laplacian is g." If[$VersionNumber < 5,Norm::"usage"="Norm[x] gives the Euclidean norm of x."] NormalD::"usage"= "NormalD[f, z] gives the outward normal derivative of f, "<> "as a function of z, with respect to the unit ball." Orthonormal::"usage"= "Orthonormal is an option for BasisH. "<> "The default value is None. The value Ball produces a "<> "basis that is orthonormal with respect to volume measure "<> "on the ball. The value Sphere produces a "<> "basis that is orthonormal with respect to surface area "<> "measure on the sphere." \!\(Partial::"\<usage\>"\ = \ \*"\"\<Partial[f, \!\(x\_j\)] gives the \ partial derivative \>\"" <> \*"\"\<of f with respect to \!\(x\_j\).\>\""\) \[CapitalPhi]::"usage"= "\[CapitalPhi][z] is the modified reflection defined in "<> "Chapter 7 of Harmonic Function Theory." PoissonKernel::"usage"="PoissonKernel[x, z] gives the Poisson kernel for the \ unit ball." PoissonKernelH::"usage"= "PoissonKernelH[x, y, t] gives the "<> "Poisson kernel for the upper half-space." Quadratic::"usage"="Quadratic is an option for Multiple "<> "in AntiLaplacian and for Region in Dirichlet." Region::"usage"="Region is an option for Dirichlet." S::"usage"="S is the south pole." Schwarz::"usage"= "Schwarz[x] gives the maximum of |u[x]|, where "<> "u ranges over all harmonic functions on the unit ball with "<> "u[0] = 0 and |u| < 1." SetDimension::"usage"="SetDimension[x, n] sets the Dimension of x to n." Singularity::"usage"= "Singularity is an option for AntiLaplacian. "<> "The default value is None." Sphere::"usage"= "Sphere is one of the values that may be assigned to the option "<> "Orthonormal." SurfaceArea::"usage"= "SurfaceArea[n] gives the surf
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