Question

\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$

Short answer

Question: Verify if the Laplace transform of the function \(f(t) = e^{at}\) is indeed \(F(s) = \frac{1}{s-a}\). Answer: Yes, the Laplace transform of the given function is indeed \(F(s) = \frac{1}{s-a}\).

Step-by-step solution

Write down the Laplace transform definition

Recall that the Laplace transform of a function \(f(t)\) is given by: $$F(s) = \int_{0}^{\infty} e^{-st}f(t)dt$$ We are given the function \(f(t) = e^{at}\) and we need to verify if its Laplace transform is indeed \(F(s) = \frac{1}{s-a}\).

Apply the Laplace transform to the given function

To find the Laplace transform of \(f(t) = e^{at}\), we need to evaluate the integral: $$F(s) = \int_{0}^{\infty} e^{-st} e^{at} dt$$ Notice that we can rewrite the integrand by combining the exponents: $$F(s) = \int_{0}^{\infty} e^{-(s-a)t} dt$$

Evaluate the integral

Since the integrand is an exponential function, we can use the formula for the integral of an exponential function: $$\int e^{\lambda x} dx = \frac{1}{\lambda}e^{\lambda x} + C$$ In our case, we have \(\lambda = -(s-a)\), so the integral becomes: $$F(s) = \frac{1}{-(s-a)}e^{-(s-a)t} \Big|_0^\infty$$

Evaluate the limits

Now, we need to evaluate the limits of the integral: \begin{align*} F(s) &= \frac{1}{-(s-a)}\left[e^{-(s-a)(\infty)} - e^{-(s-a)(0)}\right]\\ &= \frac{1}{-(s-a)}\left[0 - 1\right] \end{align*}

Simplify the expression

Finally, we can simplify the expression for \(F(s)\): $$F(s) = \frac{1}{s-a}$$ This matches the given result for the Laplace transform, so our verification is complete.

Key concepts

Exponential Functions

Exponential functions are a type of mathematical function in which a constant base is raised to a variable exponent. These functions are widely used in various fields due to their unique properties. The general form of an exponential function is given by \( f(t) = a \, e^{bt} \), where \( a \) is a constant, \( e \) is Euler's number (approximately 2.71828), and \( b \) represents the rate of growth or decay.

Exponential functions have the key feature of rapid growth or decay, making them useful in modeling phenomena like population growth, radioactive decay, and interest calculations. One crucial property of exponential functions is that their derivatives are also exponential, which makes them relatively simple to work with in calculus.

• **Rapid Growth/Decay:** The functions grow or decay at a rate proportional to their value.
• **Simple Calculus:** The rate of change of an exponential function is proportional to its value.
• **Base \( e \):** The natural exponential function, \( e^x \), is often used due to its unique derivatives and integration properties.

Improper Integrals

Improper integrals are integral expressions that involve infinite limits of integration, or integrands that approach infinity within the integration bounds. These are quite important in mathematical analysis and are frequently encountered in applications like the Laplace transform.

The Laplace transform itself makes use of an improper integral over an infinite interval starting from zero. This is crucial when handling functions that extend infinitely, for instance, in time-related problems in engineering and physics.

When evaluating improper integrals, limits are often used to handle the infinite aspects. For instance, a limit such as \[ \lim_{{b \to \infty}} \int_{0}^{b} e^{-(s-a)t} dt \] allows for the evaluation over an infinite range. Understanding and applying these concepts is essential in successfully determining outcomes in improper integrals.

• **Infinite Limits:** Used to handle infinite intervals in integration.
• **Evaluating Limits:** A necessary step to solve improper integrals analytically.

Engineering Applications

In engineering, the Laplace transform is a significant tool used to simplify the process of analyzing complex systems. Its ability to convert differential equations, which describe many engineering systems, into algebraic equations makes problem-solving more manageable.

For instance, in control engineering, the Laplace transform is employed to design and analyze systems that control processes from industrial operations to air conditioning systems. In electrical engineering, it plays a key role in circuit analysis where it helps to solve linear time-invariant systems.

• **Control Systems:** Utilized for designing systems that automatically monitor and adjust processes.
• **Circuit Analysis:** Helps to solve for current and voltage in electrical circuits by simplifying differential equations.

Physics Applications

Physics benefits immensely from the Laplace transform, especially in solving differential equations that describe physical phenomena. This includes fields like electromagnetism, optics, and vibrations. The transform simplifies the mathematics involved, making it easier to derive results that help in experimental physics and theoretical predictions.

For instance, it is extensively used in analyzing oscillations and wave equations. In quantum mechanics, it is sometimes used to compute the evolution of quantum states over time. These applications showcase the transform's versatility in translating complex physical behaviors into manageable mathematical forms.

• **Wave Equations:** Transforms complex oscillations into simpler algebraic forms.
• **Quantum Mechanics:** Aids in calculating the evolution of quantum systems.
• **Vibrations and Electromagnetism:** Solves differential equations governing these phenomena.