Calculus And Analysis
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Calculus and Analysis > Integral Transforms > General Integral Transforms
Laplace Transform The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform
(not to be confused with the Lie derivative, also commonly denoted
) is defined by (1)
where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as (2) (Oppenheim et al. 1997). The unilateral Laplace transform LaplaceTransform[f[t], t, s].
is implemented in Mathematica as
The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle). A table of several important one-sided Laplace transforms is given below. range 1
the delta function, and
In the above table, is the Heaviside step function.
is the zeroth-order Bessel function of the first kind,
The Laplace transform has many important properties. The Laplace transform existence theorem states that, if
is
is
Operation Transforms
N
F(s)
f(t),t>0 definition of a Laplace transform
1.1 y(t) inversion formula 1.2
Y(s) first derivative
1.3 second derivative 1.4 nth derivative 1.5
1.6
integration
convolution integral 1.7
F(s)G(s)
1.8 shifting in the s-plane 1.9
1.10
f(t) has period T, such that f( t + T ) = f (t)
1.11
g(t) has period T, such that g(t + T ) = - g(t) Function Transforms
N
F(s)
2.1
1
2.2
s
f(t),t>0 unit impulse at t = 0
double impulse at t = 0 2.3 unit step 2.4a u(t)
2.4b
2.5
2.6
t
2.7a
2.7b
2.8
, n=1, 2, 3,….
, k is any real number > 0 the Gamma function is given in Appendix A
2.9
2.10
2.11
2.12 2.13 2.14
2.15
2.16a 2.16b 2.17
, n=1, 2, 3,….
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29 2.30 2.31 2.32
2.33
2.34
2.35 Bessel function given in Appendix A 2.36
2.37
2.38
2.39
Modified Bessel function given in Appendix A
Laplace Transform The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits. The (unilateral) Laplace transform
(not to be confused with the Lie derivative, also commonly denoted
) is defined by (1)
where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as (2) (Oppenheim et al. 1997). The unilateral Laplace transform LaplaceTransform[f[t], t, s].
is implemented in Mathematica as
The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle). A table of several important one-sided Laplace transforms is given below. range 1
the delta function, and
In the above table, is the Heaviside step function.
is the zeroth-order Bessel function of the first kind,
The Laplace transform has many important properties. The Laplace transform existence theorem states that, if
is
is
Operation Transforms
N
F(s)
f(t),t>0 definition of a Laplace transform
1.1 y(t) inversion formula 1.2
Y(s) first derivative
1.3 second derivative 1.4 nth derivative 1.5
1.6
integration
convolution integral 1.7
F(s)G(s)
1.8 shifting in the s-plane 1.9
1.10
f(t) has period T, such that f( t + T ) = f (t)
1.11
g(t) has period T, such that g(t + T ) = - g(t) Function Transforms
N
F(s)
2.1
1
2.2
s
f(t),t>0 unit impulse at t = 0
double impulse at t = 0 2.3 unit step 2.4a u(t)
2.4b
2.5
2.6
t
2.7a
2.7b
2.8
, n=1, 2, 3,….
, k is any real number > 0 the Gamma function is given in Appendix A
2.9
2.10
2.11
2.12 2.13 2.14
2.15
2.16a 2.16b 2.17
, n=1, 2, 3,….
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29 2.30 2.31 2.32
2.33
2.34
2.35 Bessel function given in Appendix A 2.36
2.37
2.38
2.39
Modified Bessel function given in Appendix A
