Gardiolo0j

2022-09-08

Collin Gilbert

$\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\left(t\right)=1+t-\frac{8}{3}{\int }_{0}^{t}{\left(\tau -t\right)}^{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\left(\tau \right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}\tau \phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{2em}{0ex}}\stackrel{~}{\mathrm{f}}\left(s\right)={\int }_{0}^{\mathrm{\infty }}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-st}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t$

The roots of ${s}^{4}-16=0$ are given by ${s}_{n}=2\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{\mathrm{i}n\pi /2}\phantom{\rule{thinmathspace}{0ex}}$ with $n=0,1,2,3$:
${s}_{0}=2\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{1em}{0ex}}{s}_{1}=2\mathrm{i}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{1em}{0ex}}{s}_{2}=-2\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{1em}{0ex}}{s}_{3}=-2\mathrm{i}\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}\left({s}_{n}^{3}+{s}_{n}^{2}\right)\ne 0\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{1em}{0ex}}n=0,1,2,3$
With $\gamma >2$:
$\begin{array}{rl}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\left(t\right)& ={\int }_{\gamma -\mathrm{i}\mathrm{\infty }}^{\gamma +\mathrm{i}\mathrm{\infty }}\stackrel{~}{\mathrm{f}}\left(s\right)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{st}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{d}s}{2\pi \mathrm{i}}={\int }_{\gamma -\mathrm{i}\mathrm{\infty }}^{\gamma +\mathrm{i}\mathrm{\infty }}\frac{\mathrm{d}s}{2\pi \mathrm{i}}\phantom{\rule{thinmathspace}{0ex}}\frac{\left({s}^{3}+{s}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{st}\phantom{\rule{thinmathspace}{0ex}}}{{s}^{4}-16}=\sum _{n=0}^{3}\underset{s\to {s}_{n}}{lim}\frac{\left(s-{s}_{n}\right)\left({s}^{3}+{s}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{st}\phantom{\rule{thinmathspace}{0ex}}}{{s}^{4}-16}\\ & =\sum _{n=0}^{3}\frac{\left({s}_{n}^{3}+{s}_{n}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{{s}_{n}t}\phantom{\rule{thinmathspace}{0ex}}}{4{s}_{n}^{3}}=\frac{1}{4}\sum _{n=0}^{3}\left(1+\frac{1}{{s}_{n}}\right)\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{{s}_{n}t}\phantom{\rule{thinmathspace}{0ex}}=\frac{1}{4}\sum _{n=0}^{3}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{{s}_{n}t}\phantom{\rule{thinmathspace}{0ex}}+\frac{1}{4}\sum _{n=0}^{3}\frac{\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{{s}_{n}t}\phantom{\rule{thinmathspace}{0ex}}}{{s}_{n}}\\ & =\frac{1}{4}\left(\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{2t}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{2\mathrm{i}t}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-2t}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-2\mathrm{i}t}\phantom{\rule{thinmathspace}{0ex}}\right)+\frac{1}{4}\left(\frac{\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{2t}\phantom{\rule{thinmathspace}{0ex}}}{2}+\frac{\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{2\mathrm{i}t}\phantom{\rule{thinmathspace}{0ex}}}{2\mathrm{i}}+\frac{\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-2t}\phantom{\rule{thinmathspace}{0ex}}}{-2}+\frac{\phantom{\rule{thinmathspace}{0ex}}{\mathrm{e}}^{-2\mathrm{i}t}\phantom{\rule{thinmathspace}{0ex}}}{-2\mathrm{i}}\right)\\ & =\frac{1}{2}\left[\mathrm{cos}\left(2t\right)+\mathrm{cosh}\left(2t\right)\right]+\frac{1}{4}\left[\mathrm{sin}\left(2t\right)+\mathrm{sinh}\left(2t\right)\right]\end{array}$

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