       RE: Inverse Laplace output format?

• To: mathgroup at smc.vnet.net
• Subject: [mg28795] RE: [mg28769] Inverse Laplace output format?
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 13 May 2001 03:28:44 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Jose,

I love these "surgery on expressions" problems! Here is one path to the
solution.

In:=
f = (11*s^2 + 172*s + 700)/((s + 2)*(s^2 + 12*s + 100))
InverseLaplaceTransform[f, s, t]
ComplexExpand[%]
Collect[%, E^(-6*t)]
% /. a_*Cos[x_] + b_*Sin[x_] :> Sqrt[a^2 + b^2]*Cos[x - ArcTan[b/a]]
% /. ArcTan[x_] :> N[ArcTan[x]/Pi]*180*HoldForm[Degree]

Out=
(700 + 172*s + 11*s^2)/((2 + s)*(100 + 12*s + s^2))

Out=
(3 + 4*I)*E^((-6 - 8*I)*t) + (3 - 4*I)*E^((-6 + 8*I)*t) + 5/E^(2*t)

Out=
5/E^(2*t) + (6*Cos[8*t])/E^(6*t) + (8*Sin[8*t])/E^(6*t)

Out=
5/E^(2*t) + (6*Cos[8*t] + 8*Sin[8*t])/E^(6*t)

Out=
5/E^(2*t) + (10*Cos[8*t - ArcTan[4/3]])/E^(6*t)

Out=
5/E^(2*t) + (10*Cos[8*t - 53.13010235415597*HoldForm[Degree]])/E^(6*t)

The last expression looks much better in StandardForm. ArcTan[4/3] is the
exact expression for 53.13 Degree and you should usually skip the last step.

David Park

> From: Jose Lopez Jr. [mailto:chilibean2in at netzero.net]
To: mathgroup at smc.vnet.net
>
> I have the following inverse Laplace -
>
> f = (11s^2 + 172s + 700)/((s + 2)(s^2 + 12s + 100))
> InverseLaplaceTransform[f, s, t]
>
> but the answer is in exponential form.
>
> I understand the ExpToTrig command but I want to get the
> answer in "cos theta in degree" form -
>
> 5Exp[-2t] + 10Exp[-6t]cos(8t - 53.13°)
>
> How do I do that because when I use the ExpToTrig I get
>
> (3 - 4 \[ImaginaryI]) Cosh[(6 + 8 \[ImaginaryI]) t] - (3 - 4
> \[ImaginaryI])
> Sinh[(6 + 8 \[ImaginaryI]) t]
>
> and don't know what to do to get it to my "cos theta in degree" form.
>

```

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