Problem 474 - Advanced Calculus

Question

Use the Laplace transform of $$ f(t)=e^{k t} $$ where \(\mathrm{k}\) is a complex constant of the form \(\mathrm{k}=\operatorname{Re}\\{\mathrm{k}\\}+\mathrm{i} \operatorname{lm}\\{\mathrm{k}\\}\) with \(\operatorname{Re}\\{k\\}\) the real part of \(k, \operatorname{lm}\\{k\\}\) the imaginary part of \(k\), and $$ \mathrm{i} \equiv \sqrt{(-1)} $$ to find the Laplace transforms of $$ f(t)=\cosh k t, \sinh k t, \cos \mathrm{kt}, \text { and } \sin k t $$

Step-by-Step Solution

Answer

In summary, the Laplace transforms for the given functions are: 1. \(L\\{\cosh k t}\\} = \frac{2s}{s^2 - k^2}\) 2. \(L\\{\sinh k t}\\} = \frac{2k}{s^2 - k^2}\) 3. \(L\\{\cos k t}\\} = \frac{2s}{s^2 + k^2}\) 4. \(L\\{\sin k t}\\} = \frac{2k}{s^2 + k^2}\)

Step 1: Understand the basic Laplace Transform for a function

The Laplace transform of \(e^{k t}\) is given by \[L\\{e^{k t}\\}= \int_{0}^{\infty} e^{-s t}e^{k t} dt\]. This is simplified to \(L\\{e^{k t}\\}= \int_{0}^{\infty} e^{(k-s) t} dt\), and further simplified to \(L\\{e^{k t}\\}= \frac{1}{s-k}\) for \(s > \mathrm{Re}\\{k\\}\).

Step 2: Break down the hyperbolic function terms using exponential function

Hyperbolic functions could be written in terms of exponential functions, i.e., \(\cosh k t = \frac{e^{k t} + e^{-k t}}{2}\) and \(\sinh k t = \frac{e^{k t} - e^{-k t}}{2}\). By knowing the Laplace Transform of \(e^{k t}\), it is possible to find the Laplace transforms for these two functions.

Step 3: Find the Laplace transforms for the hyperbolic functions

Using the similarity principle of the Laplace Transformation, we find: \[L\\{e^{k t}+ e^{-k t}\\} = L\\{e^{k t}\\} + L\\{e^{-k t}\\}\] So, the Laplace Transform for \(\cosh k t\) becomes: \[L\\{\cosh k t}\\} = \frac{1}{s-k} + \frac{1}{s+k} = \frac{2s}{s^2 - k^2}\] Similarly, the Laplace Transform for \(\sinh k t\) becomes: \[L\\{\sinh k t}\\} = \frac{1}{s-k} - \frac{1}{s+k} = \frac{2k}{s^2 - k^2}\]

Step 4: Break down the circular function terms using Euler's formula

The circular functions can be written in terms of exponential functions using Euler's formula, i.e., \(\cos k t = \frac{e^{ikt} + e^{-ikt}}{2}\) and \(\sin k t = \frac{e^{ikt} - e^{-ikt}}{2i}\). By knowing the Laplace Transform of \(e^{ikt}\), it is possible to find the Laplace transforms for these two functions.

Step 5: Find the Laplace transforms for the circular functions

Using the similarity principle of the Laplace Transform, we can find the Laplace transforms for cosine and sine. The Laplace transform for \(\cos k t\) becomes: \[L\\{\cos k t}\\} = \frac{1}{s - ik} + \frac{1}{s + ik} = \frac{2s}{s^2 + k^2}\] Similarly, the Laplace transform for \(\sin k t\) becomes: \[L\\{\sin k t}\\} = \frac{1}{s - ik} - \frac{1}{s + ik} = \frac{-2k}{s^2 + k^2}\] Remember to divide by \(i\) as \(\sin kt = \frac{e^{i k t} - e^{-i k t}}{2i}\), so this complex number \(i\) needs to be accounted for in the transform. This results in \(L\\{\sin k t}\\} = \frac{2k}{s^2 + k^2}\] End of solution.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, much like trigonometric functions, arise in many mathematical contexts and applications. These functions are analogs of the ordinary sine and cosine functions but are based on hyperbolas rather than circles. The two primary hyperbolic functions are the hyperbolic cosine, denoted as \( \cosh(x) \), and the hyperbolic sine, denoted as \( \sinh(x) \). These functions can be expressed using exponential functions:

• Hyperbolic Cosine (\( \cosh(k t) \)): \( \cosh(k t) = \frac{e^{k t} + e^{-k t}}{2} \)
• Hyperbolic Sine (\( \sinh(k t) \)): \( \sinh(k t) = \frac{e^{k t} - e^{-k t}}{2} \)

These definitions show that hyperbolic functions are intimately tied to exponential functions, allowing them to easily transform into other forms using Laplace transforms.
Hyperbolic functions are important in calculus and complex analysis and also appear in solutions to linear differential equations and relativistic physics due to their unique properties.

Exponential Functions

Exponential functions are characterized by the constant base raised to a variable exponent. These functions play a vital role in the growth and decay processes, and are expressed generally as \( e^{x} \), where \( e \) is Euler's number, approximately \( 2.71828 \).
In the context of Laplace transforms, the exponential function \( e^{k t} \) is significant, where \( k \) is a complex number given by \( k = \text{Re}\{k\} + i\text{Im}\{k\} \). This representation splits \( k \) into its real and imaginary components:

• The real part, \( \operatorname{Re}\{k\} \), influences growth or decay rates.
• The imaginary part, \( \operatorname{Im}\{k\} \), contributes to oscillatory behavior.

Exponential functions are crucial in system analysis and control theory where they are frequently transformed using Laplace transforms to simplify the solution of differential equations.

Euler's Formula

Euler's formula provides a profound link between complex exponentials and trigonometric functions. It states that for any real number \( \theta \), the following identity holds:\[e^{i\theta} = \cos(\theta) + i\sin(\theta)\]This formula bridges the gap between the exponential function \( e^{i\theta} \) and trigonometric functions \( \cos(\theta) \) and \( \sin(\theta) \), revealing their interconnectedness.
Using Euler's formula, we can express circular (trigonometric) functions in terms of complex exponential functions:

• Cosine: \( \cos(k t) = \frac{e^{i k t} + e^{-i k t}}{2} \)
• Sine: \( \sin(k t) = \frac{e^{i k t} - e^{-i k t}}{2i} \)

These expressions are instrumental when calculating Laplace transforms for sine and cosine functions, as they utilize complex numbers to simplify the relationships and obtain solutions easily.

Complex Numbers

Complex numbers consist of a real part and an imaginary part, usually expressed as \( a + bi \). The imaginary unit \( i \) satisfies \( i^2 = -1 \). They extend the concept of one-dimensional number lines to the two-dimensional complex plane.
Each complex number can be represented uniquely in this plane, where:

• Real part, \( a \), is on the horizontal axis.
• Imaginary part, \( b \), is on the vertical axis.

Complex numbers are used in various fields such as engineering, physics, and applied mathematics for functions that encapsulate oscillations like waves and alternating currents.
The conversion of exponential functions with complex coefficients \( e^{(a+bi)t} \) provides insights into behaviors like growth, decay, and rotation, making them indispensable tools in complex function analysis and transform applications.