Calculating Laplace Transforms with the First Shifting Theorem explained

📚 Understanding the Laplace Transform and First Shifting Theorem

The Laplace transform is a powerful tool used in mathematics and engineering to solve differential equations. It transforms a function of time, $f(t)$, into a function of complex frequency, $F(s)$. The First Shifting Theorem, also known as the Frequency Shifting Theorem, simplifies the process when dealing with exponential functions multiplied by other functions.

📜 History and Background

The Laplace transform is named after Pierre-Simon Laplace, who introduced it in the 1780s. It was further developed and formalized in the 19th and 20th centuries. The transform gained popularity due to its ability to convert differential equations into algebraic equations, which are often easier to solve.

🔑 Key Principles of the First Shifting Theorem

The First Shifting Theorem states that if the Laplace transform of $f(t)$ is $F(s)$, then the Laplace transform of $e^{at}f(t)$ is $F(s-a)$. Mathematically, this is represented as:

$\mathcal{L}{e^{at}f(t)} = F(s-a)$

• 🧮 Laplace Transform Definition: The Laplace transform of a function $f(t)$ is defined as $F(s) = \int_{0}^{\infty} e^{-st} f(t) dt$.
• 📈 First Shifting Theorem: If $\mathcal{L}{f(t)} = F(s)$, then $\mathcal{L}{e^{at}f(t)} = F(s-a)$.
• ✍️ Applying the Theorem: To apply the theorem, identify $f(t)$ and $a$, find $F(s)$, and then replace $s$ with $s-a$.
• 💡 Key Idea: Multiplying a function by an exponential in the time domain corresponds to a shift in the frequency domain.

⚙️ Real-World Examples

Let's look at some examples to illustrate how to use the First Shifting Theorem:

Example 1: Find the Laplace transform of $e^{2t}t$.

1. 🔎 Identify $f(t)$ and $a$: Here, $f(t) = t$ and $a = 2$.
2. 📝 Find $F(s)$: The Laplace transform of $t$ is $\frac{1}{s^2}$.
3. ✏️ Apply the Shifting Theorem: Replace $s$ with $s-2$, so $F(s-2) = \frac{1}{(s-2)^2}$.

Therefore, $\mathcal{L}{e^{2t}t} = \frac{1}{(s-2)^2}$.

Example 2: Find the Laplace transform of $e^{-3t}\sin(2t)$.

1. 🔎 Identify $f(t)$ and $a$: Here, $f(t) = \sin(2t)$ and $a = -3$.
2. 📝 Find $F(s)$: The Laplace transform of $\sin(2t)$ is $\frac{2}{s^2 + 4}$.
3. ✏️ Apply the Shifting Theorem: Replace $s$ with $s+3$, so $F(s+3) = \frac{2}{(s+3)^2 + 4}$.

Therefore, $\mathcal{L}{e^{-3t}\sin(2t)} = \frac{2}{(s+3)^2 + 4}$.

Example 3: Find the Laplace transform of $e^{t}\cos(t)$.

1. 🔎 Identify $f(t)$ and $a$: Here, $f(t) = \cos(t)$ and $a = 1$.
2. 📝 Find $F(s)$: The Laplace transform of $\cos(t)$ is $\frac{s}{s^2 + 1}$.
3. ✏️ Apply the Shifting Theorem: Replace $s$ with $s-1$, so $F(s-1) = \frac{s-1}{(s-1)^2 + 1}$.

Therefore, $\mathcal{L}{e^{t}\cos(t)} = \frac{s-1}{(s-1)^2 + 1}$.

📝 Conclusion

The First Shifting Theorem is a valuable tool for simplifying the calculation of Laplace transforms when dealing with exponential functions. By understanding and applying this theorem, you can efficiently solve a wide range of problems in engineering and mathematics. 🎉