Complex variables and applications = 复变函数及应用 / 7th ed.

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作   者:James Ward Brown, Ruel V. Churchill著.

分类号:O174.56

ISBN:9787111133049

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简介

   本书始印于20世纪40年代,是本科数学方面为数不多的几本典范教材之一。James Ward Brown在Ruel V. Churchill原作的基础上加以扩充,编写出了复变函数课程基础教材的最新版本。    本书的首要目标是阐述复变函数应用方面的一些重要理论;另外,提供有关于残数及共形映射应用方面的介绍。    与第6版相比,第7版提供了更多的新例子、图片和补充材料;专门为讲解例子增设了许多新的小节;重新绘制了部分插图。   

目录

preface

1 complex numbers

sums and products

basic algebraic properties

further properties

moduli

complex conjugates

exponential form

products and quotients in exponential form

roots of complex numbers

examples

regions in the complex plane


2 analytic functions

functions of a complex variable

mappings

mappings by the exponential function

limits

theorems on limits

limits involving the point at infinity

.continuity

derivatives

differentiation formulas

cauchy-riemann equations

sufficient conditions for differentiability

polar coordinates

analytic functions

examples

harmonic functions

uniquely determined analytic functions

reflection principle


3 elementary functions

the exponential function

the logarithmic function

branches mid derivatives of logarithms

some identities involving logarithms

complex exponents

trigonometric functions

hyperbolic functions

inverse trigonometric and hyperbolic functions


4 integrals

derivatives of functions w(t)

definite integrals of functions w(t)

contours

contour integrals

examples

upper bounds for moduli of contour integrals

antiderivatives

examples

cauchy-goursat theorem

proof of the theorem

simply and multiply connected domains

cauchy integral formula

derivatives of analytic functions

liouville's theorem and the fundamental theorem of algebra

maximum modulus principle


5 series

convergence of sequences

convergence of series

taylor series

examples

laurent series

examples

absolute and uniform convergence of power series

continuity of sums of power series

integration and differentiation of power series

uniqueness of series representations

multiplication and division of power series6 residues and poles


6 residues

cauchy's residue theorem

using a single residue

the three types of isolated singular points

residues at poles

examples

zeros of analytic functions

zeros and poles

behavior of f near isolated singular points


7 applications of residues

evaluation of improper integrals

example

improper integrals from fourier analysis

jordan's lemma

indented paths

an indentation around a branch point

integration along a branch cut

definite integrals involving sines and cosines

argument principle

rouche's theorem

inverse laplace transforms

examples


8 mapping by elementary functions

linear transformations

the transformation w=1/z

mappings by 1/z

linear fractional transformations

an implicit form

mappings of the upper half plane

the transformation w=sin z

mappings by z2 and branches of z1/2

square roots of polynomials

riemann surfaces

surfaces for related functions


9 conformal mapping

preservation of angles

scale factors

local inverses

harmonic conjugates

transformations of harmonic functions

transformations of boundary conditions


10 applications of conformal mapping

steady temperatures

steady temperatures in a half plane

a related problem

temperatures in a quadrant

electrostatic potential

potential in a cylindrical space

two-dimensional fluid flow

the stream function

flows around a corner and around a cylinder


11 the schwarz-christoffel transformation

mapping the real axis onto a polygon

schwarz-christoffel transformation

triangles and rectangles

degenerate polygons

fluid flow in a channel through a slit

flow in a channel with an offset

electrostatic potential about an edge of a conducting plate


12 integral formulas of the poisson type

poisson integral formula

dirichlet problem for a disk

related boundary value problems

schwarz integral formula

dirichlet problem for a half plane

neumann problems

appendixes

bibliography

table of transformations of regions

index


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