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October 13th, 2013, 09:19 AM   #1
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Inverse laplace transform

Hi, I am trying to do the inverse Laplace transform for this equation I arrived at with partial fractions. Two of the terms have complex numbers in them and I see that they are simplified to Cos and Sin terms from their exponential form (the process is quoted as 'Eulers identity'). Could somebody explain how this process is carried out?

L^-1 : 5*(-1+j)/(s+j500)

and 5*(-1-j)/(s+j500)

The inverse of these I have as (-1+j)e^(j500*t) and (-1-j)e^(-j500*t) respectively (I hope that is correct!)

Where my total equation is equal to these two terms and one other non-complex exponential term added together (it is for current in a resistor inductor circuit with a sinusoidal source voltage).

Many thanks,

Ryan
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October 13th, 2013, 06:23 PM   #2
jks
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Re: Inverse laplace transform

Hi Rydog21,

I believe that you need to use Euler's Formula:



I also think that there is a mis-statement in your problem. The reason is that the denominators are both the same but the signs of the exponents in the Inverse Laplace Transform are opposite. I will assume that you meant to find:



Given that:

we get:

[1]

By Euler's formula, we can write as and

we can write as

Substituting these into the right hand side of equation [1] we get:





Using Euler's Formula again, this reduces to:



Since the imaginary terms cancel and the real terms add giving:



Note, of course, that could be replaced with
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October 15th, 2013, 10:35 AM   #3
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Re: Inverse laplace transform

Thanks for that, I understand now. You're the man.
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