Laplace transform

Dec 2015
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Find the recurrence relation of \(\displaystyle \mathcal{L}\{ \sin^{n} (t) \} \; , n\in \mathbb{N}\).
 
Dec 2015
973
128
Earth
\(\displaystyle \mathcal{L} \{f^{(n)}\}=\mathcal{L} \{ \sin(x+n\pi /2 ) \} = s^n\mathcal{L} \{f\}-s^{n-1}f(0)-s^{n-2}f'(0)-s^{n-3} f''(0)-\ldots -f^{(n-1)}(0)\).

How to define \(\displaystyle f^{(n)}(0)\) ? , it can take more than two values.
\(\displaystyle \mathcal{L} \{ \sin(x+n\pi /2 ) \}=s^{n}\mathcal{L}\{f \}-\sum_{i=2}^{n}s^{n-i}f^{(i-1)}(0) \).
 
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romsek

Math Team
Sep 2015
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Find the recurrence relation of \(\displaystyle \mathcal{L}\{ \sin^{n} (t) \} \; , n\in \mathbb{N}\).
given your second post, what does $\sin^n(x)$ mean?

$\sin^3(x) = \sin(x) \cdot \sin(x) \cdot \sin(x)~ ?$

$\sin^3(x) = \dfrac{d^3}{dx^3}\sin(x)~ ?$
 
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Dec 2015
973
128
Earth
sin\(\displaystyle ^{n}\) - exponent ; sin\(\displaystyle ^{(n)} \)-derivative.
f=f(t).
 
Dec 2015
973
128
Earth
For example , \(\displaystyle n=2\).
\(\displaystyle \mathcal{L} ( sin^2 t )=\dfrac{1}{s}\cdot \mathcal{L} ( 2sin2t ) =\dfrac{4}{s(4+s^2 ) }\).