Glossary

7.1 Definition and Properties of Laplace Transforms

3 min read•august 6, 2024

Laplace transforms are powerful tools for solving differential equations. They convert complex time-domain problems into simpler algebraic equations in the transform domain, making them easier to solve.

This section covers the definition, properties, and applications of Laplace transforms. We'll learn about linearity, shifting properties, and how to handle periodic functions and calculus operations using these transforms.

Definition and Properties

Laplace Transform Fundamentals

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  • Laplace transform converts a function f(t)f(t) from the time domain to the transform domain, denoted as F(s)F(s)
  • Defined by the improper integral L{f(t)}=F(s)=∫0∞f(t)e−stdt\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} f(t)e^{-st} dt, where ss is a complex variable
  • Time domain represents the original function with respect to time tt, while the transform domain represents the function with respect to the complex variable ss
  • Existence of the Laplace transform depends on the function f(t)f(t) and its behavior as tt approaches infinity

Linearity Property

  • states that the Laplace transform is a linear operator
  • For constants aa and bb and functions f(t)f(t) and g(t)g(t), L{af(t)+bg(t)}=aL{f(t)}+bL{g(t)}\mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} + b\mathcal{L}\{g(t)\}
  • Allows for the Laplace transform of a sum of functions to be computed as the sum of their individual Laplace transforms
  • Enables the use of superposition to solve linear differential equations using Laplace transforms

Shifting and Periodic Functions

First Shifting Property

  • First shifting property, also known as the time-shift property, relates the Laplace transform of a shifted function to its original Laplace transform
  • For a function f(t)f(t) with Laplace transform F(s)F(s), the Laplace transform of f(t−a)u(t−a)f(t-a)u(t-a) is given by e−asF(s)e^{-as}F(s), where u(t−a)u(t-a) is the shifted by aa
  • Shifting a function in the time domain results in a multiplication by an exponential term in the transform domain
  • Useful for solving differential equations with initial conditions or analyzing systems with time delays

Second Shifting Property

  • Second shifting property, also known as the frequency-shift property, relates the Laplace transform of a function multiplied by an exponential term to its original Laplace transform
  • For a function f(t)f(t) with Laplace transform F(s)F(s), the Laplace transform of eatf(t)e^{at}f(t) is given by F(s−a)F(s-a)
  • Multiplying a function by an exponential term in the time domain results in a shift of the complex variable ss in the transform domain
  • Helps in analyzing systems with exponential growth or decay, or in solving differential equations with exponential terms

Transform of Periodic Functions

  • Periodic functions are functions that repeat their values at regular intervals, such as sin⁡(t)\sin(t) and cos⁡(t)\cos(t) with period 2π2\pi
  • Laplace transform of a periodic function f(t)f(t) with period TT is given by L{f(t)}=11−e−sT∫0Tf(t)e−stdt\mathcal{L}\{f(t)\} = \frac{1}{1-e^{-sT}}\int_0^T f(t)e^{-st} dt
  • The integral is taken over one period of the function, and the result is divided by 1−e−sT1-e^{-sT} to account for the infinite repetition of the function
  • Enables the analysis of systems with periodic inputs or behaviors, such as electrical circuits with alternating current (AC) sources or mechanical systems with oscillating components

Calculus Properties

Derivative Property

  • Derivative property relates the Laplace transform of the derivative of a function to its original Laplace transform
  • For a function f(t)f(t) with Laplace transform F(s)F(s), the Laplace transform of its derivative f′(t)f'(t) is given by L{f′(t)}=sF(s)−f(0)\mathcal{L}\{f'(t)\} = sF(s) - f(0), where f(0)f(0) is the initial value of f(t)f(t)
  • The Laplace transform of the second derivative f′′(t)f''(t) is given by L{f′′(t)}=s2F(s)−sf(0)−f′(0)\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0), and this pattern continues for higher-order derivatives
  • Simplifies the process of solving differential equations by transforming them into algebraic equations in the transform domain

Integral Property

  • Integral property relates the Laplace transform of the integral of a function to its original Laplace transform
  • For a function f(t)f(t) with Laplace transform F(s)F(s), the Laplace transform of its integral ∫0tf(τ)dτ\int_0^t f(\tau) d\tau is given by L{∫0tf(τ)dτ}=1sF(s)\mathcal{L}\left\{\int_0^t f(\tau) d\tau\right\} = \frac{1}{s}F(s)
  • The Laplace transform "inverts" the process of integration, making it easier to solve integral equations or find the response of a system to an input
  • Helps in analyzing the behavior of systems described by integro-differential equations, such as control systems with feedback or electrical circuits with capacitors

Key Terms to Review (14)

Absolute convergence: Absolute convergence refers to the property of a series where the series of absolute values converges. This means that if you take a series of numbers and replace each term with its absolute value, and that new series converges, then the original series is said to converge absolutely. This concept is crucial because absolute convergence guarantees the convergence of the original series, which allows for certain mathematical manipulations and applications.
First Shifting Theorem: The First Shifting Theorem is a fundamental property of Laplace transforms that states if you shift a function in the time domain by a constant, its Laplace transform is multiplied by an exponential factor in the s-domain. Specifically, if you have a function $$f(t)$$ and you shift it to $$f(t - a)$$ for $$t \geq a$$, then the Laplace transform of this shifted function relates to the original by the equation $$\mathcal{L}\{f(t - a)u(t - a)\} = e^{-as}F(s)$$, where $$F(s)$$ is the Laplace transform of $$f(t)$$. This theorem is important for analyzing systems in engineering and physics where delays or shifts are present.
Impulse Function: The impulse function, also known as the Dirac delta function, is a mathematical construct that represents an idealized point source or unit impulse at a specific time. It is characterized by being zero everywhere except at a single point, where it integrates to one, making it useful in systems analysis, particularly in the context of Laplace transforms to model instantaneous changes in systems.
Inverse laplace transform: The inverse Laplace transform is a mathematical operation that takes a function defined in the Laplace domain and transforms it back into the time domain. This operation is crucial for solving differential equations, as it allows one to find the original function or solution after working with its transformed version. Understanding the inverse Laplace transform is essential for applying Laplace transforms to initial value problems and leveraging their properties for efficient computation.
L{e^at}: l{e^at} represents the Laplace transform of the exponential function $$e^{at}$$, where $$a$$ is a constant and $$t$$ is the time variable. This transform converts a function of time into a function of a complex variable, providing insights into system behavior in engineering and physics. The formula is useful in solving differential equations and analyzing linear time-invariant systems.
L{sin(ωt)}: The expression l{sin(ωt)} represents the Laplace transform of the sine function, specifically sin(ωt), where ω is a constant frequency. This transform is a powerful tool used to analyze linear time-invariant systems and can convert differential equations in the time domain into algebraic equations in the complex frequency domain. Understanding this transformation is essential for solving ordinary differential equations with sine functions as inputs or responses.
Linearity property: The linearity property is a fundamental characteristic of linear operators, stating that the Laplace transform of a linear combination of functions is equal to the same linear combination of their respective Laplace transforms. This property simplifies the process of transforming complex functions into the Laplace domain, allowing for easier manipulation and analysis of differential equations.
Region of convergence: The region of convergence refers to the set of values in the complex plane for which a given Laplace transform converges to a finite value. It is crucial because it determines the stability and behavior of the system being analyzed, indicating where the integral that defines the Laplace transform is well-defined. Understanding this region is essential for solving differential equations and analyzing dynamic systems using Laplace transforms.
S-domain: The s-domain is a complex frequency domain used in the analysis of linear time-invariant systems through Laplace transforms. It allows us to convert differential equations into algebraic equations, making them easier to manipulate and solve. By transforming functions from the time domain into the s-domain, we can analyze system behavior, stability, and response characteristics efficiently.
Second Shifting Theorem: The Second Shifting Theorem is a property of the Laplace transform that allows for the transformation of a shifted function in time to be expressed in terms of the original function. It states that if you have a function $f(t)$ and you apply a shift in time by $a$, the Laplace transform of $f(t - a)u(t - a)$ can be computed using the transform of $f(t)$, where $u(t - a)$ is the unit step function. This theorem is essential for solving differential equations with initial conditions and analyzing systems affected by time delays.
Solving initial value problems: Solving initial value problems involves finding a specific solution to a differential equation that satisfies given initial conditions at a particular point. These initial conditions typically include the value of the function and possibly its derivatives at a specified time, which helps determine a unique solution from the general family of solutions to the differential equation. This process is crucial when using techniques like Laplace transforms, as it allows for straightforward handling of both ordinary and partial differential equations.
System Response Analysis: System response analysis refers to the study of how a dynamic system reacts to external inputs over time, particularly focusing on the system's output based on given initial conditions and inputs. This concept is crucial in understanding how systems behave in response to various forces and how they stabilize or oscillate after disturbances, especially when using tools like Laplace transforms to solve differential equations.
T-domain: The t-domain refers to the time domain in which functions or signals are defined and analyzed, primarily concerning time-dependent variables. In the context of Laplace transforms, it is the realm where functions are represented as a function of time, typically denoted as t. Understanding the t-domain is crucial as it serves as the foundation for transforming these time-dependent functions into the frequency domain, facilitating easier manipulation and analysis in differential equations.
Unit step function: The unit step function, often denoted as $u(t)$, is a piecewise function that is zero for negative values of time and one for positive values, effectively 'turning on' at $t=0$. It is crucial in engineering and mathematics for modeling sudden changes in systems, such as a switch being turned on or off. The unit step function serves as a foundation for understanding discontinuous forcing functions and plays a significant role in the context of Laplace transforms, which are used to analyze linear time-invariant systems.
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