Riley hobson bence · Problem 13.28

Integral transforms

Mathematics — Methods · laplace-transforms-periodic-functions-heaviside

Problem

Show that the Laplace transform of

f(t-a)H(t-a)

, where

a \geq 0

, is

e^{-as}\bar{f}(s)

and that, if

g(t)

is a periodic function of period

T

\bar{g}(s)

can be written as

\frac{1}{1-e^{-sT}}\int_0^T e^{-st}g(t)\,dt.

(a) Sketch the periodic function defined in

0 \leq t \leq T

g(t) = \begin{cases} 2t/T & 0 \leq t < T/2, \\ 2(1-t/T) & T/2 \leq t \leq T, \end{cases}

and, using the previous result, find its Laplace transform. (b) Show, by sketching it, that

\frac{2}{T}\left[tH(t) + 2\sum_{n=1}^{\infty}(-1)^n\left(t-\tfrac{1}{2}nT\right)H\!\left(t-\tfrac{1}{2}nT\right)\right]

is another representation of

g(t)

and hence derive the relationship

\tanh x = 1 + 2\sum_{n=1}^{\infty}(-1)^n e^{-2nx}.