# Laplace transform

#### idontknow

Find the recurrence relation of $$\displaystyle \mathcal{L}\{ \sin^{n} (t) \} \; , n\in \mathbb{N}$$.

#### idontknow

$$\displaystyle \mathcal{L} \{f^{(n)}\}=\mathcal{L} \{ \sin(x+n\pi /2 ) \} = s^n\mathcal{L} \{f\}-s^{n-1}f(0)-s^{n-2}f'(0)-s^{n-3} f''(0)-\ldots -f^{(n-1)}(0)$$.

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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1 person

#### romsek

Math Team
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?

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

$\sin^3(x) = \dfrac{d^3}{dx^3}\sin(x)~ ?$

2 people

#### idontknow

sin$$\displaystyle ^{n}$$ - exponent ; sin$$\displaystyle ^{(n)}$$-derivative.
f=f(t).

#### romsek

Math Team
sin$$\displaystyle ^{n}$$ - exponent ; sin$$\displaystyle ^{(n)}$$-derivative.
f=f(t).
then what does post 2 have to do with post 1?

1 person

#### idontknow

For example , $$\displaystyle n=2$$.
$$\displaystyle \mathcal{L} ( sin^2 t )=\dfrac{1}{s}\cdot \mathcal{L} ( 2sin2t ) =\dfrac{4}{s(4+s^2 ) }$$.