Complex variables

CONTENTS
21.5
Inhomogeneous problems – Green’s functions
751
Similarities to Green’s functions for ordinary differential equations; general
boundary-value problems; Dirichlet problems; Neumann problems
21.6
21.7
Exercises
Hints and answers
767
773
22
22.1
22.2
Calculus of variations
The Euler–Lagrange equation
Special cases
775
776
777
F does not contain y explicitly; F does not contain x explicitly
22.3
Some extensions
781
Several dependent variables; several independent variables; higher-order
derivatives; variable end-points
22.4
22.5
Constrained variation
Physical variational principles
785
787
Fermat’s principle in optics; Hamilton’s principle in mechanics
22.6
22.7
22.8
22.9
22.10
General eigenvalue problems
Estimation of eigenvalues and eigenfunctions
Adjustment of parameters
Exercises
Hints and answers
790
792
795
797
801
23
23.1
23.2
23.3
23.4
Integral equations
Obtaining an integral equation from a differential equation
Types of integral equation
Operator notation and the existence of solutions
Closed-form solutions
803
803
804
805
806
Separable kernels; integral transform methods; differentiation
23.5
23.6
23.7
23.8
23.9
Neumann series
Fredholm theory
Schmidt–Hilbert theory
Exercises
Hints and answers
813
815
816
819
823
24
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
Complex variables
Functions of a complex variable
The Cauchy–Riemann relations
Power series in a complex variable
Some elementary functions
Multivalued functions and branch cuts
Singularities and zeros of complex functions
Conformal transformations
Complex integrals
824
825
827
830
832
835
837
839
845
xiii