Last Updated: 10/5/26

Laplace transform

Definition

Laplace Cards

Page 1 t-domain v1.4

Page 1 s-domain v1.4

Page 2 t-domain v1.4

Page 2 s-domain v1.4

Step functions

Piecewise function defs

Example

Transform pairs

Table of basic transforms

Impulse function

Step function

Sine

Example

Example 2

Identities:

List

Multiplication by constant

Addition

Differentiation

Integration

Delay

Frequency shift

Scaling time

Multiplication by time

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Inverse transform

^†Definition

Partial fractions

Forms

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Matching forms

Example 1

Complex roots

Example 1

Poles and zeros

Pole location vs t form

Superposition

Cancellation

RLC circuit

Ex

Plot of location

Example 1

Example 2

Example 3

Poles and zeros (cont.)

H(jω) frequency response

Ex 1

|H(jω)| from distance

Ex 1

∠H(jω) from angles

Ex 1

Spirule

Initial/final value theorems

Initial value theorem

Final value theorem

Example

Example 2

Example 3

Circuits

s-domain circuit elements

R L C

Initial Conditions

Example

s-domain solutions

Example

Example 2

Example 3

Example 4

t-domain waveforms

Example

Example 2

Example 3

Example 4

Example 5

Example 6

Impulse function

Example

Basis functions

Sinusoids as eigenfunctions