Using the integral definition for , we obtain. Explore Solution We actually show that the integral defining equals the formula for values of s with and that the extension to other values of s is inferred by our knowledge about the domain of a rational function. We can use the property of linearity to find new Laplace transforms from known transforms. Show that. Extra Example 1. Explore Solution for Extra Example 1.
A direct approach using the definition is tedious. Recall that can be written as the linear combination. Using the linearity of the Laplace transform, we have. Inverting the Laplace transform is usually accomplished with the aid of a table of known Laplace transforms and the technique of partial fraction expansion.
Find the inverse Laplace transform. Using linearity and lines 6 and 7 of Table We will now investigate explore formula Definition of the Inverse Laplace Transform. This is the situation we will consider.
The inverse Laplace Transform is defined with a contour integral This integral is called the Bromwich integral and sometimes it is called the Fourier-Mellin integral. We can use the Residue Calculus to evaluate the Bromwich integral. The details are left for the reader to investigate. Integration along a Curve in the s-Plane 1. Integration around a Pole 1. Integration around a Path Not Containing a Pole 1.
Residues 1. Integration around Two or More Poles in the s-Plane 1. The Fourier Series and Integral 2.
The Fourier Series 2. Exponential Form of the Fourier Series 2. The Fourier Integral 2. The Unit Step Function 2. Convergence Factors 2. The Complex Fourier Integral Transform 2. The Laplace Transformation 3. Introduction 3.
Transforms of Constants 3. The Laplace Transform of Exponentials 3. The Laplace Transform of Imaginary Exponents 3. The Laplace Transform of Trigonometric Terms 3. The Laplace Transform of Hyperbolic Functions 3. The Laplace Transform of Complex Exponentials 3. Transforms of More Complicated Functions 3. Additional Practice with Sine Waves 3. The Laplace Transform of a Derivative 3. The Inverse Laplace Transformation 4. Introduction 4. Functions of s from Electronic Networks 4. Functions of s Involving Simple Poles 4. Laplace Transform Theorems 5. Introduction 5. Linear s-Plane Translation 5.
Final Value Theorem 5. Initial Value Theorem 5. Real Translation 5. Complex Differentiation 5. Complex Integration 5. Sectioning a Function of Time 5. The Convolution Theorem 5. Scale Change Theorem 5. Network Analysts by Means of the Laplace Transformation 6. Introduction 6. Relay Damping Problems 6.
The Wien-Bridge Oscillator 6. A Phase-Shift Oscillator 6. Odd and Even Functions of s 6. R-C Voltage Step-up Networks 6. Active Integrating and Differentiating Networks 6. Condition: Good. A copy that has been read, but remains in clean condition.
The spine may show signs of wear. Pages can include limited notes and highlighting, and the copy can include previous owner inscriptions. Seller Inventory GI3N More information about this seller Contact this seller 2. More information about this seller Contact this seller 3. More information about this seller Contact this seller 4. More information about this seller Contact this seller 5. More information about this seller Contact this seller 6. More information about this seller Contact this seller 7. Published by Dover Publications About this Item: Dover Publications, Condition: Used: Good.
More information about this seller Contact this seller 8. Soft cover. Condition: Very Good. ISBN: The theory of ordinary differential equations in real and complex domains is here clearly explained and analyzed. Not only classical theory, but also the main developments of modern times are covered. Seller Inventory More information about this seller Contact this seller 9.
More information about this seller Contact this seller About this Item: N. XVII, pp. Published by Dover Publications Inc. About this Item: Dover Publications Inc. Condition: New.
New edition. Language: English. Brand new Book. Sevart, Department of Mechanical Engineering, University of Wichita"An extremely useful textbook for both formal classes and for self-study. This text is designed to remedy that need by supplying graduate engineering students especially electrical engineering with a course in the basic theory of complex variables, which in turn is essential to the understanding of transform theory.
Presupposing a good knowledge of calculus, the book deals lucidly and rigorously with important mathematical concepts, striking an ideal balance between purely mathematical treatments that are too general for the engineer, and books of applied engineering which may fail to stress significant mathematical ideas.
Now consider differentiation. After a chapter on additional properties of the Laplace integral, the book ends with four chapters on the application of transform theory to the solution of ordinary linear integrodifferential equations with constant coefficients, impulse functions, periodic functions, and the increasingly important Z transform. Never used!. Don't have an account? Contact the MathWorld Team.
The text is divided into two basic parts: The first part Chapters is devoted to the theory of complex variables and begins with an outline of the structure of system analysis and an explanation of basic mathematical and engineering terms. Chapter 2 treats the foundation of the theory of a complex variable, centered around the Cauchy-Riemann equations. The next three chapters -- conformal mapping, complex integration, and infinite series -- lead up to a particularly important chapter on multivalued functions, explaining the concepts of stability, branch points, and riemann surfaces.