1. Complete Laplace Transform Table

Basic Properties and Theorems

Linearity

First Shifting Theorem (Frequency Shift)

Second Shifting Theorem (Time Shift)

Scaling Theorem

Differentiation (Time Domain)

Integration (Time Domain)

Differentiation (Frequency Domain)

Convolution Theorem

where

Initial Value Theorem

Final Value Theorem

(if limit exists)

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Standard Laplace Transform Pairs

		Conditions
(Dirac delta)
(Unit step)

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Additional Common Functions

(pulse)

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Partial Fraction Decomposition (for Inverse Transforms)

When inverting rational functions, use partial fractions:

Case 1: Distinct Linear Factors

Case 2: Repeated Linear Factors

Case 3: Quadratic Factors

Complete the square:

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Common Inverse Transform Patterns

Pattern 1:

Pattern 2:

Pattern 3:

Pattern 4:

Pattern 5:

Pattern 6: