Difference between revisions of "Laplace transform"

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$$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} e^{-zt}f(t) dt.$$
 
$$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} e^{-zt}f(t) dt.$$
  
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=Examples=
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# Find a function $f$ such that $\mathscr{L}\{f\}(z)=f(z)$. <br />
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<strong>Solution:</strong> We know that if $g(t)=t^{\ell}$ and $\mathrm{Re}(\ell) > 1$, then $\mathscr{L}\{g\}(z)=\dfrac{\Gamma(\ell+1)}{z^{\ell+1}}$, where $\Gamma$ denotes the [[gamma function]]. We will seek a function $f$ with the property that $\mathscr{L}\{f\}(z)=f(z)$. Let $s \in \mathbb{C}$ with $0<\mathrm{Re}(s)<1$ be as of yet undetermined. Suppose $f$ has the form
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$$f(t)=\sqrt{\Gamma(s)}t^{-s}+\sqrt{\Gamma(1-s)}t^{s-1}.$$
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Then we may compute
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$$\begin{array}{ll}
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\mathscr{L}\{f\}(z)&=\sqrt{\Gamma(s)} \dfrac{\Gamma(1-s)}{z^{-s+1}} + \sqrt{\Gamma(1-s)} \dfrac{\Gamma(s)}{z^s} \\
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&= \sqrt{\Gamma(s)\Gamma(1-s)} \left[ \dfrac{\sqrt{\Gamma(s)}\Gamma(1-s)}{z^{-s+1}\sqrt{\Gamma(s)\Gamma(1-s)}} +  \dfrac{\sqrt{\Gamma(1-s)}\Gamma(s)}{z^s\sqrt{\Gamma(s)\Gamma(1-s)}} \right] \\
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&= \sqrt{\Gamma(s)\Gamma(1-s)} \left[ \dfrac{\sqrt{\Gamma(1-s)}}{z^{-s+1}} +  \dfrac{\sqrt{\Gamma(s)}}{z^s} \right] \\
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&= \sqrt{\Gamma(s)\Gamma(1-s)} \left[ \sqrt{\Gamma(1-s)}z^{s-1} + \sqrt{\Gamma(s)}z^{-s} \right] \\
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&= \sqrt{\Gamma(s)\Gamma(1-s)} f(z).
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\end{array}.$$
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If we pick $s$ so that $\sqrt{\Gamma(1-s)}z^{s-1} + \sqrt{\Gamma(s)}=1$ we will have found the function $f$. By properties of the gamma function we know that $\Gamma(1-z)\Gamma(z)=\dfrac{\pi}{\sin(\pi z)}$ and so we see that we must pick $s$ so that $\dfrac{\pi}{\sin(\pi s)}=1,$ i.e., $s = \dfrac{\arcsin(\pi)}{\pi},$
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which [http://www.wolframalpha.com/input/?i=arcsin%28pi%29%2Fpi yields] $s=\dfrac{\pi}{2} - i\log(\pi + \sqrt{\pi^2-1})$.
 
=Videos=
 
=Videos=
 
[https://www.youtube.com/watch?v=u3v6V7SXrl8 Laplace transform of power function with real exponent] <br />
 
[https://www.youtube.com/watch?v=u3v6V7SXrl8 Laplace transform of power function with real exponent] <br />
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[https://www.youtube.com/watch?v=BAme-njI8sE Laplace transform of cosine integral]<br />
 
[https://www.youtube.com/watch?v=BAme-njI8sE Laplace transform of cosine integral]<br />
 
[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br />
 
[https://www.youtube.com/watch?v=TppV_yDY3EQ Laplace transform of exponential integral]<br />
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=References=
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[http://math.stackexchange.com/questions/150390/laplace-transform-identity Laplace transform identity]

Revision as of 23:06, 16 March 2015

Let $f \colon \mathbb{R} \rightarrow \mathbb{C}$ be a function, then the Laplace transform of $f$ is the function defined by $$\mathscr{L}\{f\}(z) = \displaystyle\int_0^{\infty} e^{-zt}f(t) dt.$$

Examples

  1. Find a function $f$ such that $\mathscr{L}\{f\}(z)=f(z)$.

Solution: We know that if $g(t)=t^{\ell}$ and $\mathrm{Re}(\ell) > 1$, then $\mathscr{L}\{g\}(z)=\dfrac{\Gamma(\ell+1)}{z^{\ell+1}}$, where $\Gamma$ denotes the gamma function. We will seek a function $f$ with the property that $\mathscr{L}\{f\}(z)=f(z)$. Let $s \in \mathbb{C}$ with $0<\mathrm{Re}(s)<1$ be as of yet undetermined. Suppose $f$ has the form $$f(t)=\sqrt{\Gamma(s)}t^{-s}+\sqrt{\Gamma(1-s)}t^{s-1}.$$ Then we may compute $$\begin{array}{ll} \mathscr{L}\{f\}(z)&=\sqrt{\Gamma(s)} \dfrac{\Gamma(1-s)}{z^{-s+1}} + \sqrt{\Gamma(1-s)} \dfrac{\Gamma(s)}{z^s} \\ &= \sqrt{\Gamma(s)\Gamma(1-s)} \left[ \dfrac{\sqrt{\Gamma(s)}\Gamma(1-s)}{z^{-s+1}\sqrt{\Gamma(s)\Gamma(1-s)}} + \dfrac{\sqrt{\Gamma(1-s)}\Gamma(s)}{z^s\sqrt{\Gamma(s)\Gamma(1-s)}} \right] \\ &= \sqrt{\Gamma(s)\Gamma(1-s)} \left[ \dfrac{\sqrt{\Gamma(1-s)}}{z^{-s+1}} + \dfrac{\sqrt{\Gamma(s)}}{z^s} \right] \\ &= \sqrt{\Gamma(s)\Gamma(1-s)} \left[ \sqrt{\Gamma(1-s)}z^{s-1} + \sqrt{\Gamma(s)}z^{-s} \right] \\ &= \sqrt{\Gamma(s)\Gamma(1-s)} f(z). \end{array}.$$ If we pick $s$ so that $\sqrt{\Gamma(1-s)}z^{s-1} + \sqrt{\Gamma(s)}=1$ we will have found the function $f$. By properties of the gamma function we know that $\Gamma(1-z)\Gamma(z)=\dfrac{\pi}{\sin(\pi z)}$ and so we see that we must pick $s$ so that $\dfrac{\pi}{\sin(\pi s)}=1,$ i.e., $s = \dfrac{\arcsin(\pi)}{\pi},$ which yields $s=\dfrac{\pi}{2} - i\log(\pi + \sqrt{\pi^2-1})$.

Videos

Laplace transform of power function with real exponent
Laplace transform of $\sin(\sqrt{t})$
Laplace transform of impulse function
Laplace transform of sine integral
Laplace transform of cosine integral
Laplace transform of exponential integral

References

Laplace transform identity