Laplace transform of complex exponential

In summary, the conversation discusses the use of Laplace transform to find the transform of a complex number. It is confirmed that the transform of a complex number is still 1/(s-complex_number), as every number can be represented as a complex number. The use of direct integration is also mentioned as a method to demonstrate this. The conversation also mentions the use of Laplace transform to find the transforms of sine and cosine functions.
  • #1
EvLer
458
0
I just want to be sure I understand this correctly, usually L[f(t)] = 1/(s-a), where [tex]f(t) = e^{at}[/tex], but if it is a complex number would still be 1/(s - complex_number)?
techinically, i think it should be, since every number can be reprsented as complex number. Just want to be sure about this with Laplace transform.
thanks much.
 
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  • #2
Definitely! You can demonstrate it by direct integration.
 
  • #3
EvLer said:
I just want to be sure I understand this correctly, usually L[f(t)] = 1/(s-a), where [tex]f(t) = e^{at}[/tex], but if it is a complex number would still be 1/(s - complex_number)?
techinically, i think it should be, since every number can be reprsented as complex number. Just want to be sure about this with Laplace transform.
thanks much.
Yes and it is a nice way to find the laplace transforms of sin and cos.
L[cos(a x)+i sin(a x)]=L[exp(i x)]=1/(s-a i)=(s+a i)/(s^2+a^2)
hence (equating real and imaginary parts)
L[cos(a x)]=s/(s^2+a^2)
L[sin(a x)]=a/(s^2+a^2)
 

FAQ: Laplace transform of complex exponential

What is a Laplace transform?

A Laplace transform is a mathematical operation that converts a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems in the frequency domain.

What is a complex exponential?

A complex exponential is a complex number expressed in the form z = rejθ, where r is the magnitude and θ is the angle in the complex plane. It can also be written as z = a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

How do you take the Laplace transform of a complex exponential?

To take the Laplace transform of a complex exponential, you can use the formula L{eat} = 1/(s-a), where s is the complex frequency and a is the real part of the complex exponential. If the complex exponential is in the form ejωt, then the Laplace transform is L{ejωt} = 1/(s-jω).

What is the inverse Laplace transform of a complex exponential?

The inverse Laplace transform of a complex exponential is given by the formula L-1{1/(s-a)} = eat. It can also be written as L-1{1/(s-jω)} = ejωt. This means that the original function in the time domain is an exponential function with a real or imaginary coefficient, depending on the form of the complex exponential.

Why is the Laplace transform of a complex exponential useful?

The Laplace transform of a complex exponential is useful because it allows us to analyze systems and functions in the frequency domain, which can provide insights and solutions that may not be easily obtainable in the time domain. It also simplifies differential equations and makes it easier to solve them using algebraic methods.

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