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Index

T-periodic function, 26 integral equation, 88
L, 6 integration theorem, 24, 25
−1
L , 7, 54 inverse Laplace transform, 7, 24, 29
L, 12 linearity, 30

absolute convergence, 9, 12 jump discontinuity at a point, 10

boundary-value problem, 76 Laplace equation, 90
Bromwich line, 54 Laplace transform, 6
convergence, 6, 9
complementary error function, 71
divergence, 6
complex inversion formula, 54
existence, 12
convolution, 44
linearity, 15
property, 45
notation, 6
Convolution Theorem, 45
periodic function, 27

Derivative Theorem, 20, 22 properties, 15
Dirac delta function, 49, 51 Laplace transformation, 7

elliptic equation, 89 ordinary differential equation, 71, 85

error function, 48
parabolic equation, 89
Euler formula, 7
partial differential equation, 89
exponential order, 12
partial fraction decomposition, 32

First Expansion Theorem, 56 Partial Fraction Expansion Theorem, 38
First Shifting Theorem, 17 piecewise continuity, 11
First Translation Theorem, 17
Second Shifting Theorem, 19
Fourier transform, 54
Second Translation Theorem, 19
Fourier–Mellin formula, 54
sine integral, 25
general solution, 75 system of differential equations, 79

heat equation, 89 uniform convergence, 10, 13
Heaviside (step) function, 30 unit step function, 30
Heaviside Expansion Theorem, 43
wave equation, 90, 98
hyperbolic equation, 89

impulse response, 49
initial value problem, 71

104