Glossary

11.2 Inverse Laplace Transforms

3 min read•july 30, 2024

Inverse Laplace transforms are the key to unlocking solutions in the time domain. They let us take complex functions in the s-domain and bring them back to familiar territory, revealing how systems behave over time.

This powerful tool builds on what we've learned about Laplace transforms. By reversing the process, we can solve tricky differential equations and analyze real-world systems with ease. It's like having a secret decoder ring for math!

Inverse Laplace Transform

Definition and Uniqueness

Top images from around the web for Definition and Uniqueness

  • differential equations - How to solve laplace inverse using convolution - Mathematics Stack Exchange View original

    Is this image relevant?

  • Laplace inverse transform of the complex expression - Mathematics Stack Exchange View original

    Is this image relevant?

  • differential equations - How to solve laplace inverse using convolution - Mathematics Stack Exchange View original

    Is this image relevant?

1 of 3

Top images from around the web for Definition and Uniqueness

  • differential equations - How to solve laplace inverse using convolution - Mathematics Stack Exchange View original

    Is this image relevant?

  • Laplace inverse transform of the complex expression - Mathematics Stack Exchange View original

    Is this image relevant?

  • differential equations - How to solve laplace inverse using convolution - Mathematics Stack Exchange View original

    Is this image relevant?

1 of 3
  • maps function F(s) in complex s-domain back to function f(t) in time domain denoted as L−1{F(s)}L^{-1}\{F(s)\}
  • Unique correspondence exists between F(s) and f(t) ensuring one-to-one mapping
  • Rational function F(s) has inverse Laplace transform when numerator degree < denominator degree
  • provides theoretical basis for computing inverse transforms (rarely used in practice due to complexity)

Key Properties

  • property allows inverse transform of linear combinations L−1{aF(s)+bG(s)}=aL−1{F(s)}+bL−1{G(s)}L^{-1}\{aF(s) + bG(s)\} = aL^{-1}\{F(s)\} + bL^{-1}\{G(s)\} (a and b are constants)
  • relates product of transforms to convolution of functions L−1{F(s)G(s)}=f(t)∗g(t)L^{-1}\{F(s)G(s)\} = f(t) * g(t) (* denotes convolution)
  • relates limit of F(s) as s approaches 0 to limit of f(t) as t approaches infinity
  • relates limit of sF(s) as s approaches infinity to limit of f(t) as t approaches 0

Partial Fraction Decomposition

Decomposition Process

  • Breaks down complex rational functions into simpler fractions with known inverse Laplace transforms
  • Expresses F(s) as sum of simpler fractions based on roots of denominator polynomial
  • Applies to proper rational functions (numerator degree < denominator degree)
  • Distinct linear factors yield terms of form As−a\frac{A}{s-a} (A is constant to be determined)
  • Repeated linear factors yield terms of form A(s−a)n\frac{A}{(s-a)^n} (n is multiplicity of root)
  • Quadratic factors yield terms of form As+Bs2+bs+c\frac{As+B}{s^2+bs+c} (A and B are constants to be determined)

Coefficient Determination

  • Coefficients in partial fraction expansion found using method of equating coefficients or cover-up method
  • Equating coefficients involves setting up system of equations by comparing expanded form to original function
  • Cover-up method involves multiplying both sides by denominator factor and evaluating at root to find coefficient
  • For quadratic factors, simultaneous equations solved to determine A and B

Inverse Laplace Transform Properties

Shifting and Scaling

  • transforms exponential factors in s-domain L−1{F(s−a)}=eatf(t)L^{-1}\{F(s-a)\} = e^{at}f(t) (a is real constant)
  • manipulates time scale L−1{F(as)}=1∣a∣f(ta)L^{-1}\{F(as)\} = \frac{1}{|a|}f(\frac{t}{a}) (a is non-zero constant)
  • Combination of shifting and scaling applies to more complex functions L−1{F(as+b)}L^{-1}\{F(as+b)\}
  • (time shift) L−1{e−asF(s)}=f(t−a)u(t−a)L^{-1}\{e^{-as}F(s)\} = f(t-a)u(t-a) (u(t) is unit step function)
  • (frequency shift) L−1{F(s+a)}=e−atf(t)L^{-1}\{F(s+a)\} = e^{-at}f(t)

Applications

  • Simplifies process of finding inverse Laplace transforms for complex functions
  • Useful in solving differential equations (converting to algebraic equations)
  • Facilitates analysis of (transfer functions and system responses)
  • Enables manipulation of time delays and frequency shifts in

Inverse Laplace Transform Tables

Common Functions

  • Exponential functions (1s+a→e−at\frac{1}{s+a} \rightarrow e^{-at})
  • Trigonometric functions (ss2+ω2→cos⁡(ωt)\frac{s}{s^2+\omega^2} \rightarrow \cos(\omega t))
  • Hyperbolic functions (ωs2−ω2→sinh⁡(ωt)\frac{\omega}{s^2-\omega^2} \rightarrow \sinh(\omega t))
  • Rational functions (1s2→t\frac{1}{s^2} \rightarrow t)
  • Step functions (1s→u(t)\frac{1}{s} \rightarrow u(t))
  • Ramp functions (1s2→tu(t)\frac{1}{s^2} \rightarrow tu(t))
  • Impulse functions and derivatives (1→δ(t)1 \rightarrow \delta(t))

Advanced Functions

  • Bessel functions (used in cylindrical systems)
  • Error functions (used in heat transfer and diffusion problems)
  • Combinations of basic functions (s(s2+a2)(s2+b2)→1a2−b2[cos⁡(bt)−cos⁡(at)]\frac{s}{(s^2+a^2)(s^2+b^2)} \rightarrow \frac{1}{a^2-b^2}[\cos(bt)-\cos(at)])

Effective Table Usage

  • Recognize need for function manipulation to match table entries
  • Apply linearity property to break down complex functions into simpler components
  • Utilize shifting and scaling properties to transform functions into standard forms
  • Combine multiple table entries to solve more complex problems
  • Practice identifying patterns and relationships between s-domain and time-domain functions

Key Terms to Review (23)

Augustin-Louis Cauchy: Augustin-Louis Cauchy was a French mathematician whose work laid the foundations for many areas of mathematics, including analysis and differential equations. He is especially known for his contributions to the theory of functions, the concept of convergence, and the study of differential equations, which are crucial in understanding physical systems. His methods and techniques continue to influence modern mathematics, particularly in relation to inverse transforms and special types of equations.
Bromwich Integral Formula: The Bromwich Integral Formula is a method used to compute inverse Laplace transforms, allowing us to recover a time-domain function from its Laplace transform. This formula is essential in transforming complex frequency domain representations back into their corresponding time domain forms, making it a vital tool in solving differential equations and analyzing linear systems.
Coefficient determination: Coefficient determination, often represented as $R^2$, is a statistical measure that explains the proportion of variance in the dependent variable that can be predicted from the independent variable(s) in a regression model. This measure is critical in assessing the goodness-of-fit of a model, indicating how well the independent variables explain the variability of the dependent variable. Higher values of $R^2$ suggest a better fit and more reliable predictions.
Control Systems: Control systems are mathematical models used to analyze and design systems that regulate themselves, ensuring desired outputs based on given inputs. These systems can be found in various applications, from engineering to biology, where they play a crucial role in maintaining stability and achieving specific performance criteria under varying conditions. Understanding control systems involves studying their behavior in response to disturbances and how they maintain equilibrium through feedback mechanisms.
Convolution Theorem: The convolution theorem states that the Laplace transform of the convolution of two functions is equal to the product of their individual Laplace transforms. This fundamental property connects the time domain operations of convolution with the frequency domain operations represented by Laplace transforms, making it a powerful tool for analyzing linear systems, especially when dealing with differential equations and system responses.
Dirac Delta Function: The Dirac delta function is a mathematical construct that represents an idealized point mass or point charge, defined such that it is zero everywhere except at one point, where it is infinitely high and integrates to one. This function is particularly useful in applications such as inverse Laplace transforms, where it acts as a tool for simplifying the representation of impulses or instantaneous events in mathematical models.
Existence Theorem: An existence theorem is a fundamental result in mathematics that guarantees the existence of solutions to certain types of problems, particularly in the context of differential equations. These theorems typically establish conditions under which a solution can be assured, often related to continuity and differentiability of functions involved. They are crucial for understanding when initial value problems and boundary value problems can be solved effectively.
Final Value Theorem: The final value theorem provides a method for determining the steady-state behavior of a system as time approaches infinity, using the Laplace transform. This theorem states that if a function is stable and has a limit as time goes to infinity, then the final value can be computed from its Laplace transform. This connects to properties of Laplace transforms, the process of finding inverse transforms, and is particularly useful in solving differential equations, allowing for quick insights into long-term system behavior without directly solving the equations.
First Shifting Theorem: The First Shifting Theorem is a property of the Laplace transform that states if you have a function $f(t)$ and you shift it by a constant 'a', the Laplace transform of the shifted function is related to the original function by a simple exponential factor. This theorem is essential for solving differential equations, as it allows us to incorporate initial conditions easily and can simplify the process of finding solutions in terms of the Laplace transform.
Heaviside Step Function: The Heaviside step function, often denoted as H(t), is a piecewise function that equals 0 for all negative input values and 1 for all non-negative input values. It is widely used in mathematics and engineering to model systems that switch on or off at a specific point in time, serving as a foundational component in the analysis of differential equations and the inverse Laplace transform.
Initial Value Problems: Initial value problems (IVPs) are a type of differential equation that require the solution to satisfy specific conditions at a given point, usually the starting point in time. These conditions typically involve specifying the value of the function and possibly its derivatives at that point. IVPs are crucial when applying methods like Laplace transforms, as they allow us to find unique solutions to differential equations by imposing these constraints.
Initial Value Theorem: The Initial Value Theorem states that the value of a function at time zero can be determined from its Laplace transform. Specifically, if a function $$f(t)$$ has a Laplace transform $$F(s)$$, then the initial value of $$f(t)$$ at $$t=0$$ can be found as $$f(0) = ext{lim}_{s \to \infty} sF(s)$$. This theorem is crucial for analyzing systems and solving differential equations in the context of transforms.
Inverse laplace transform: The inverse Laplace transform is a mathematical operation that takes a function defined in the Laplace domain and converts it back into the time domain. This process is essential in solving differential equations and analyzing dynamic systems, allowing for the determination of time-dependent behavior from frequency-domain representations. Understanding the inverse Laplace transform is critical for applying properties of Laplace transforms and for finding solutions to problems in engineering and physics.
Inverse transform formulas: Inverse transform formulas are mathematical expressions used to convert a function from a transformed domain back to its original domain. In the context of Laplace transforms, these formulas provide a way to retrieve a time-domain function from its corresponding Laplace-transformed function, enabling the analysis and solution of differential equations.
Laplace Transform Table: A Laplace transform table is a comprehensive reference that lists common functions alongside their Laplace transforms, providing an efficient way to convert time-domain functions into the frequency domain. This table simplifies the process of solving differential equations and analyzing systems by offering quick access to transforms and their properties, enhancing problem-solving efficiency and accuracy.
Linearity: Linearity refers to a property of mathematical functions and transformations where they satisfy two key conditions: additivity and homogeneity. This means that if you have two inputs, the output of the function for the sum of those inputs is the same as the sum of the outputs for each input individually, and if you scale an input by a factor, the output is scaled by the same factor. This principle is foundational in understanding various mathematical concepts like transformations, differential equations, and systems, linking them through their predictable behavior.
Ordinary differential equations: Ordinary differential equations (ODEs) are mathematical equations that relate a function to its derivatives, representing how a quantity changes with respect to one independent variable. ODEs play a crucial role in modeling real-world phenomena in various fields, particularly in understanding dynamic systems and processes, which can be analyzed using techniques like Laplace transforms and applications in engineering and physics.
Partial fraction decomposition: Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions, making it easier to integrate or apply inverse transformations. This method is particularly useful when dealing with rational expressions that have polynomial numerators and denominators, allowing for the manipulation of these expressions into a form that can be more easily analyzed in the context of Laplace transforms and their inverses.
Pierre-Simon Laplace: Pierre-Simon Laplace was a French mathematician and astronomer known for his significant contributions to statistics, celestial mechanics, and the development of the Laplace transform. His work laid the foundation for many areas in mathematics and physics, particularly in understanding dynamic systems and solving differential equations through the use of his transforms.
Scaling Property: The scaling property refers to a fundamental aspect of the inverse Laplace transform, which states that if a function is scaled by a constant factor in the time domain, the corresponding scaling factor will appear in the frequency domain. This means that if a function is multiplied by a constant, its inverse Laplace transform will reflect this change by multiplying the transform by the same constant. Understanding this property allows for easier manipulation of transforms when dealing with functions that require scaling adjustments.
Second Shifting Theorem: The Second Shifting Theorem is a property of the Laplace Transform that allows for the shifting of a function in the time domain to be reflected in the s-domain. This theorem is crucial because it simplifies the process of finding inverse Laplace transforms for functions that include a step function, enabling easier analysis of systems with delayed responses.
Shifting Property: The shifting property is a fundamental concept in the context of inverse Laplace transforms, which allows for the manipulation of functions through time shifts. This property states that if you have a function multiplied by an exponential decay term, it can be shifted in the time domain by modifying its Laplace transform accordingly. Understanding this property is crucial for solving differential equations and analyzing systems since it helps relate time-shifted signals back to their original forms.
Signal Processing: Signal processing involves the analysis, interpretation, and manipulation of signals to enhance their quality or extract useful information. It plays a crucial role in various applications such as telecommunications, audio processing, and image analysis, enabling clearer communication and better data representation.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide