Transform Methods in Mathematics

2.

The Laplace Transform

2.1.

Definition and Existence Theory

2.1.1.

One-Sided Laplace Transform

2.1.1.1.

Definition for Causal Functions

2.1.1.2.

Transform Variable s

2.1.2.

Two-Sided Laplace Transform

2.1.2.1.

Definition for Bilateral Functions

2.1.2.2.

Relationship to One-Sided Transform

2.1.3.

Existence Conditions

2.1.3.1.

Exponential Order

2.1.3.2.

Piecewise Continuity

2.1.3.3.

Absolute Integrability Conditions

2.2.

Region of Convergence

2.2.1.

Definition in the s-Plane

2.2.2.

ROC for Different Signal Types

2.2.2.1.

Right-Sided Signals

2.2.2.2.

Left-Sided Signals

2.2.2.3.

Two-Sided Signals

2.2.2.4.

Finite Duration Signals

2.2.3.

Properties of the ROC

2.2.3.1.

2.2.3.2.

Pole-Zero Relationship

2.2.4.

ROC and System Properties

2.2.4.1.

2.2.4.2.

Stability (BIBO)

2.3.

Properties of the Laplace Transform

2.3.1.

Linearity Property

2.3.2.

Time Shifting (Delay)

2.3.3.

Frequency Shifting (s-Domain Translation)

2.3.4.

2.3.5.

Time Differentiation

2.3.6.

Time Integration

2.3.7.

Frequency Differentiation

2.3.8.

Convolution in Time Domain

2.3.9.

Multiplication by Powers of t

2.3.10.

Initial Value Theorem

2.3.11.

Final Value Theorem

2.4.

Transform Pairs and Tables

2.4.1.

Elementary Functions

2.4.1.1.

Unit Step Function

2.4.1.2.

Unit Impulse Function

2.4.1.3.

Exponential Functions

2.4.1.4.

Power Functions

2.4.1.5.

Sinusoidal Functions

2.4.1.6.

Hyperbolic Functions

2.4.2.

Composite Functions

2.4.2.1.

Exponentially Modulated Functions

2.4.2.2.

Piecewise Functions

2.4.2.3.

Periodic Functions

2.5.

Inverse Laplace Transform

2.5.1.

Partial Fraction Decomposition

2.5.1.1.

2.5.1.2.

2.5.1.3.

Complex Conjugate Poles

2.5.2.

Bromwich Inversion Integral

2.5.2.1.

Contour Selection

2.5.2.2.

Residue Calculation

2.5.2.3.

Branch Cuts and Singularities

2.5.3.

Convolution Method

2.5.4.

Table Lookup Method

2.6.

2.6.1.

Ordinary Differential Equations

2.6.1.1.

Initial Value Problems

2.6.1.2.

Higher Order ODEs

2.6.1.3.

Systems of ODEs

2.6.2.

Circuit Analysis

2.6.2.1.

Resistor-Inductor-Capacitor Circuits

2.6.2.2.

Transfer Functions

2.6.2.3.

Transient Analysis

2.6.3.

Control Systems

2.6.3.1.

System Transfer Functions

2.6.3.2.

Stability Analysis

2.6.3.3.

Controller Design

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1. Foundations of Transform Methods

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3. Fourier Analysis