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Laplace Transform of e^(t/2)
Published on October 27, 2024
Calculus
The Laplace transform of e^(t/2) is 1/(s - 1/2), provided s > 1/2.
Question
laplace transform e^{t/2}
Answer
To find the Laplace transform of
Definition: The Laplace transform of a function
where
Applying the definition to our problem:
-
Substitute the function:
We need to find, so we substitute into the definition: -
Simplify the integrand:
Combine the exponential terms: -
Evaluate the integral:
This is now a standard integral of the form. Applying this, we get: -
Apply the limits of integration:
- Upper limit: As
approaches infinity, the term will approach zero if the exponent is negative. This requires that , or . - Lower limit: At
, the exponential term becomes 1.
Therefore, we have: - Upper limit: As
Final answer:
The Laplace transform of