102–106. Laplace transforms 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) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:
F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).
Verify the following Laplace transforms, where a is a real number.
106. f(t) = cos(at) → F(s) = s/(s² + a²)

Verified step by step guidance

1. Start with the definition of the Laplace transform for the function \(f(t) = \cos(at)\): \[F(s) = \int_0^{\infty} e^{-st} \cos(at) \, dt\]
2. Recall that the integral of the product of an exponential and a cosine function can be evaluated using the formula: \[\int_0^{\infty} e^{-pt} \cos(qt) \, dt = \frac{p}{p^2 + q^2}\] where \(p > 0\) and \(q\) are real numbers.
3. In this problem, identify \(p = s\) and \(q = a\), since the integral matches the form with \(e^{-st}\) and \(\cos(at)\).
4. Apply the formula directly to get: \[F(s) = \frac{s}{s^2 + a^2}\] which is the Laplace transform of \(\cos(at)\).
5. To fully verify, you could also perform the integral by expressing \(\cos(at)\) as the real part of \(e^{iat}\) and then integrating \(e^{-(s - ia)t}\), ensuring the integral converges for \(s > 0\).

Key Concepts

Definition of the Laplace Transform

The Laplace transform converts a time-domain function f(t) into a complex frequency-domain function F(s) using the integral F(s) = ∫₀^∞ e^(-st) f(t) dt. This transformation simplifies solving differential equations by turning them into algebraic equations, where s is a complex variable with a positive real part to ensure convergence.

Improper Integrals and Convergence

Evaluating the Laplace transform involves improper integrals extending to infinity. Understanding when these integrals converge is crucial, typically requiring the exponential term e^(-st) to dominate f(t) as t → ∞. This ensures the integral has a finite value, allowing the transform to exist and be computed.

Laplace Transform of Trigonometric Functions

The Laplace transform of functions like cos(at) is derived using integration techniques or known formulas, resulting in F(s) = s / (s² + a²). This formula is essential for solving differential equations involving oscillatory behavior, linking time-domain cosine functions to rational expressions in s.