📊 Laplace Transforms

Comprehensive Formula Handbook for Differential Equations

📐 Basic Laplace Transform Definition

Definition: The Laplace Transform of a function \(f(t)\) is defined as:

$$L\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) \, dt$$

where \(s\) is a complex number, \(s = \sigma + j\omega\)

Region of Convergence (ROC): The transform exists for values of \(s\) where the integral converges, typically \(\text{Re}(s) > a\) for some real number \(a\).

⚡ Properties of Laplace Transform

1. Linearity Property
$$A f_1(t) + B f_2(t) \leftrightarrow A F_1(s) + B F_2(s)$$
2. Time Shifting Property
$$f(t - a)u(t - a) \leftrightarrow e^{-as} F(s)$$

where \(u(t)\) is the unit step function

3. Frequency Shifting Property
$$e^{at} f(t) \leftrightarrow F(s - a)$$
4. Differentiation in Time
$$f'(t) \leftrightarrow sF(s) - f(0)$$ $$f''(t) \leftrightarrow s^2F(s) - sf(0) - f'(0)$$ $$f^{(n)}(t) \leftrightarrow s^nF(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)$$
5. Integration Property
$$\int_0^t f(\lambda) \, d\lambda \leftrightarrow \frac{1}{s} F(s)$$
6. Multiplication by Time
$$t^n f(t) \leftrightarrow (-1)^n \frac{d^n}{ds^n} F(s)$$
7. Division by Time
$$\frac{f(t)}{t} \leftrightarrow \int_s^\infty F(\sigma) \, d\sigma$$
8. Time Scaling Property
$$f(at) \leftrightarrow \frac{1}{a} F\left(\frac{s}{a}\right), \quad a > 0$$
9. Convolution Theorem
$$f(t) * g(t) \leftrightarrow F(s) \cdot G(s)$$

where \(f*g = \int_0^t f(\tau)g(t-\tau) \, d\tau\)

10. Differentiation in Frequency
$$t f(t) \leftrightarrow -\frac{d}{ds} F(s)$$

📋 Laplace Transform Table

No. \(f(t)\) \(F(s) = L\{f(t)\}\)
1 \(\delta(t)\) (unit impulse) \(1\)
2 \(u(t)\) (unit step) \(\frac{1}{s}\)
3 \(t\) \(\frac{1}{s^2}\)
4 \(t^n\), \(n = 1,2,3,\ldots\) \(\frac{n!}{s^{n+1}}\)
5 \(t^p\), \(p > -1\) \(\frac{\Gamma(p+1)}{s^{p+1}}\)
6 \(\sqrt{t}\) \(\frac{\sqrt{\pi}}{2s^{3/2}}\)
7 \(e^{at}\) \(\frac{1}{s - a}\)
8 \(t e^{at}\) \(\frac{1}{(s-a)^2}\)
9 \(t^n e^{at}\) \(\frac{n!}{(s-a)^{n+1}}\)
10 \(\sin(at)\) \(\frac{a}{s^2 + a^2}\)
11 \(\cos(at)\) \(\frac{s}{s^2 + a^2}\)
12 \(t \sin(at)\) \(\frac{2as}{(s^2 + a^2)^2}\)
13 \(t \cos(at)\) \(\frac{s^2 - a^2}{(s^2 + a^2)^2}\)
14 \(\sin(at + b)\) \(\frac{s\sin(b) + a\cos(b)}{s^2 + a^2}\)
15 \(\cos(at + b)\) \(\frac{s\cos(b) - a\sin(b)}{s^2 + a^2}\)
16 \(e^{at} \sin(bt)\) \(\frac{b}{(s-a)^2 + b^2}\)
17 \(e^{at} \cos(bt)\) \(\frac{s-a}{(s-a)^2 + b^2}\)
18 \(\sinh(at)\) \(\frac{a}{s^2 - a^2}\)
19 \(\cosh(at)\) \(\frac{s}{s^2 - a^2}\)
20 \(e^{at} \sinh(bt)\) \(\frac{b}{(s-a)^2 - b^2}\)
21 \(e^{at} \cosh(bt)\) \(\frac{s-a}{(s-a)^2 - b^2}\)

🔄 Inverse Laplace Transform

Definition: The Inverse Laplace Transform is given by:

$$f(t) = L^{-1}\{F(s)\} = \frac{1}{2\pi j} \int_{\gamma - j\infty}^{\gamma + j\infty} e^{st} F(s) \, ds$$

where \(\gamma\) is a real number chosen so that all singularities of \(F(s)\) lie to the left of the line \(\text{Re}(s) = \gamma\)

Practical Methods for Finding Inverse Transforms:

  • Partial Fraction Decomposition
  • Convolution Theorem
  • Residue Theorem (Complex Analysis)
  • Transform tables and linearity

🎯 Important Theorems

Initial Value Theorem:

$$\lim_{t \to 0^+} f(t) = \lim_{s \to \infty} s F(s)$$

Provided the limit exists

Final Value Theorem:

$$\lim_{t \to \infty} f(t) = \lim_{s \to 0} s F(s)$$

Valid if \(sF(s)\) has no poles in the right half-plane or on the imaginary axis (except possibly at the origin)

Convolution Theorem:

$$L\{f(t) * g(t)\} = F(s) \cdot G(s)$$ $$L^{-1}\{F(s) \cdot G(s)\} = f(t) * g(t) = \int_0^t f(\tau) g(t - \tau) \, d\tau$$

🔬 Applications

1. Solving Differential Equations: Convert ODEs to algebraic equations in the s-domain

2. Circuit Analysis: Analyze RLC circuits, transfer functions, and frequency response

3. Control Systems: Stability analysis, transfer functions, and system response

4. Signal Processing: Filter design, system identification, and signal analysis

5. Mechanical Systems: Vibration analysis, spring-mass-damper systems

💡 Pro Tip: When solving differential equations using Laplace transforms:

  1. Take the Laplace transform of both sides
  2. Apply initial conditions
  3. Solve the algebraic equation for \(F(s)\)
  4. Use partial fractions if necessary
  5. Take the inverse Laplace transform to get \(f(t)\)