The Laplace Transform of (f(t) 3^t): A Comprehensive Guide

In mathematics, the Laplace transform is a powerful tool used to solve differential equations and analyze systems in engineering and physics. A specific instance of its application involves determining the Laplace transform of the exponential function 3^t. This article will walk you through the process step-by-step and provide a detailed derivation of the Laplace transform of (f(t) 3^t).

Step-By-Step Derivation of the Laplace Transform of (f(t) 3^t)

To find the Laplace transform of the function (f(t) 3^t), we start from the definition of the Laplace transform:

mathcal{L}{f(t)} int_0^infty e^{-st} f(t) dt

Substituting f(t) 3^t:

mathcal{L}{3^t} int_0^infty e^{-st} cdot 3^t dt

This expression can be rewritten as:

mathcal{L}{3^t} int_0^infty e^{-s - ln(3)t} dt

Integral Evaluation

The integral of e^{-alpha t} from 0 to infinity is given by:

int_0^infty e^{-alpha t} dt frac{1}{alpha} quad text{for } alpha > 0

In our case, we have alpha s - ln(3). Therefore, for the integral to converge, we must have:

s > ln(3)

Evaluation and Final Answer

Using the integral formula, we can evaluate:

mathcal{L}{3^t} frac{1}{s - ln(3)} quad text{for } s > ln(3)

Summary

In conclusion, the Laplace transform of the function (f(t) 3^t) is:

mathcal{L}{3^t} frac{1}{s - ln(3)}

This result is valid for values of s greater than (ln(3)).

Additional Contexts and Applications

The Laplace transform of (3^t) has various applications in fields such as engineering, physics, and control theory. For example, it can be used to solve initial value problems involving exponential functions. Understanding how to derive and use Laplace transforms is crucial for students and professionals working in these areas.

Keywords: Laplace transform, (3^t), mathematical analysis