Sine transform

Eqn 1

Use complex notation for the sin

Eqn 2

Note that with

Eqn 3

So that

Eqn 4

In the usual way let

Eqn 5

The trap rule estimate of the integrals is then


Eqn 6

Note that there are no – signs on the alternate steps.  This is because the sum naturally starts from zero rather than from negative infinity.