Thanks to Euler’s formula (page 168) and the algebraic

QUESTION:

Thanks to Euler’s formula (page 168) and the algebraic properties of complex numbers, several of the entries of Table 7.1 can be derived from a single formula; namely,

(6) \(\mathscr{L}\left\{e^{a+i b) t}\right\}(s)=\frac{s-a+i b}{(s-a)^{2}+b^{2}}, S>a\)

(a) By computing the integral in the definition of the Laplace transform on page 353 with \(f(t)=e^{(a+i b) t}\), show that

\(\mathscr{L}\left\{e^{(a+i b) t}\right\}(s)=\frac{1}{s-(a+i b)}, s>a\)

(b) Deduce (6) from part (a) by showing that

\(\frac{1}{s-(a+i b)}=\frac{s-a+i b}{(s-a)^{2}+b^{2}}\)

(c) By equating the real and imaginary parts in formula (6), deduce the last two entries in Table 7.1.

Equation transcription:

Text transcription:

{L}{e^{a+i b) t}}(s)=frac{s-a+i b}{(s-a)^{2}+b^{2}}, S>a

f(t)=e^{(a+i b) t}

r{L}{e^{(a+i b) t}}(s)=frac{1}{s-(a+i b)}, s>a

frac{1}{s-(a+i b)}=frac{s-a+i b}{(s-a)^{2}+b^{2}}