Chapter 18Green's Functions
Green's functions are among the most versatile tools in applied mathematics. They provide a powerful alternative to solving differential equations. They are also very useful in transforming differential equations into integral equations, which are preferred in certain cases like the scattering problems. Propagator interpretation of the Green's functions is also very useful in quantum field theory, and with their path integral representation, they are the starting point of modern perturbation theory. In this chapter, we introduce the basic features of both the time-dependent and the time-independent Green's functions, which have found a wide range of applications in science and engineering. We also introduce the time-independent perturbation theory.
18.1 Time-Independent Green's Functions in One Dimension
We start with the differential equation
where 
is the Sturm–Liouville operator:
with 
and 
as continuous functions defined in the interval . Along ...
Get Mathematical Methods in Science and Engineering, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
