Mathematical Methods in the Physical Sciences · Problem 2.5

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · separation-of-variables-solutions

Problem

Show that the solutions of (2.5) can also be written as

X = \begin{cases} e^{ikx}, \\ e^{-ikx}, \end{cases} \qquad Y = \begin{cases} \sinh ky, \\ \cosh ky. \end{cases}

Also show that these solutions are equivalent to (2.7) if

k

is real and equivalent to (2.18) if

k

is pure imaginary. (See Chapter 2, Section 12.) Also show that

X = \sin k(x-a)

Y = \sinh k(y-b)

are solutions of (2.5).