- #1
SqueeSpleen
- 141
- 5
I have two questions:
I had to find the Laplace transform of:
[itex]t \cdot sin(t)[/itex]
Not by definition, using a table of transforms and the properties.
I did:
[itex]sin(t) = i \cdot sinh(it) = i \cdot \frac{e^{t}}{2}-i \cdot \frac{e^{-t}}{2} [/itex]
Then
[itex]t \cdot sin(t) = it \cdot \frac{e^{t}}{2}-it \cdot \frac{e^{-t}}{2}[/itex]
And
[itex]t^{n}e^{-at}=\frac{n!}{(s+a)^{n+1}}[/itex]
So
[itex]\frac{i}{2} t^{1}e^{-it}=\frac{i}{2} \frac{1}{(s+i)^{2}}=\frac{i}{2} \frac{1}{s^{2}+2si-1}[/itex] and [itex]\frac{i}{2} t^{1}e^{it}=\frac{i}{2} \frac{1}{(s-i)^{2}}=\frac{i}{2} \frac{1}{s^{2}-2si-1}[/itex]
[itex]\frac{i}{2} \frac{s^{2}-2si-1-(s^{2}+2si-1)}{(s^2+1)^2}=\frac{2s}{(s^2+1)^2}[/itex]
I want to know other way of calculate this without using the definition, because I don't know if I'm meant to use complex numbers.
The other problem is:
[itex]f(t)=\frac{sin(t)}{t}[/itex] if [itex]t \neq 0[/itex]
[itex]f(t)=1[/itex] if [itex]t = 0[/itex]
Find the Mac Laurin serie of the function and check that [itex]L\left\{f(t)\right\}=arctan(\frac{1}{s})[/itex] s >1
I find the following serie:
[itex]\sum_{n=0}^{\infty} (-1)^n (\frac{1}{s})^{2n+1}\frac{1}{2n+1}[/itex]
But it diverges for s>1, so it's useless to my purpose.
And I had problems trying to compute a Taylor Serie of [itex]arctan(\frac{1}{s})[/itex], I don't know what point to use.
Sorry if I made some gramatical mistakes, I don't speak English very well.
I had to find the Laplace transform of:
[itex]t \cdot sin(t)[/itex]
Not by definition, using a table of transforms and the properties.
I did:
[itex]sin(t) = i \cdot sinh(it) = i \cdot \frac{e^{t}}{2}-i \cdot \frac{e^{-t}}{2} [/itex]
Then
[itex]t \cdot sin(t) = it \cdot \frac{e^{t}}{2}-it \cdot \frac{e^{-t}}{2}[/itex]
And
[itex]t^{n}e^{-at}=\frac{n!}{(s+a)^{n+1}}[/itex]
So
[itex]\frac{i}{2} t^{1}e^{-it}=\frac{i}{2} \frac{1}{(s+i)^{2}}=\frac{i}{2} \frac{1}{s^{2}+2si-1}[/itex] and [itex]\frac{i}{2} t^{1}e^{it}=\frac{i}{2} \frac{1}{(s-i)^{2}}=\frac{i}{2} \frac{1}{s^{2}-2si-1}[/itex]
[itex]\frac{i}{2} \frac{s^{2}-2si-1-(s^{2}+2si-1)}{(s^2+1)^2}=\frac{2s}{(s^2+1)^2}[/itex]
I want to know other way of calculate this without using the definition, because I don't know if I'm meant to use complex numbers.
The other problem is:
[itex]f(t)=\frac{sin(t)}{t}[/itex] if [itex]t \neq 0[/itex]
[itex]f(t)=1[/itex] if [itex]t = 0[/itex]
Find the Mac Laurin serie of the function and check that [itex]L\left\{f(t)\right\}=arctan(\frac{1}{s})[/itex] s >1
I find the following serie:
[itex]\sum_{n=0}^{\infty} (-1)^n (\frac{1}{s})^{2n+1}\frac{1}{2n+1}[/itex]
But it diverges for s>1, so it's useless to my purpose.
And I had problems trying to compute a Taylor Serie of [itex]arctan(\frac{1}{s})[/itex], I don't know what point to use.
Sorry if I made some gramatical mistakes, I don't speak English very well.
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