University Mathematics Handbook (2015)
XIII. Fourier Series and Integral Transforms
Chapter 3. Laplace Transformation Formulas
3.1 Definition
a. Let be a piecewise continuous function : →. Function of real variable is calledLaplace transform of .
b. If there exist real constants and such that , then is defined for every .
3.2 Formulas of Laplace Transform
a. Linearity
b. Differential formula:
c. n-th order differential:
d.
e.
f.
g. If is a periodic function with period , then
3.3 Heaviside Step Function
For all real positive , the function is Heaviside step function.
a.
b. If is Laplace transform of and , then .
c. If , then .
3.4 Dirac Delta Function
a. Let be a given real number. "Function" , holding for every function continuous in the neighborhood of , and for all set containing a neighborhood of , is called the Dirac delta function on .
b. only exists as a function of functions. One common description of is a "limit of a process". For every , let us define function .
Properties of :
1.
2. , for every
3. , for every
Function can be perceived as equal to the so-called limit .
c.
d.
3.5 Convolution (see 2.3 h,j)
a.
b.
c. If there exist constants , , and such that and for every , then
, and there holds
.
3.6 Table of Laplace Transforms
are real numbers and
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2 |
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3 |
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4 |
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(See V.3.3) |
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10 |
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11 |
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12 |
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14 |
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15 |
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16 |
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17 |
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18 |
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19 |
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(see XI.3.5) |
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26 |