تحويل لابلاس: الفرق بين النسختين
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سطر 21:
: <math>f(t) = \mathcal{L}^{-1} \{F\} = \mathcal{L}^{-1}_s \{F(s)\} := \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{ \gamma - i T}^{ \gamma + i T} e^{st} F(s)\,ds,</math>
==خصائص ونظريات==
هناك مجموعة من الخصائص لتحويل لابلاس لابد من معرفتها لتسهيل استخدامه وبخاصة في تحليل النظم الخطية، من أهمها حالات التفاضل والتكامل.
والجدول التالي يبين ملخصا لهذه الخصائص والنظريات:
إذا كان هناك دالتين:
''f''(''t'') و ''g''(''t'')
وكان تحويل لابلاس لهما هو:
''F''(''s'') و ''G''(''s'')
: <math> f(t) = \mathcal{L}^{-1} \{ F(s) \} </math>
: <math> g(t) = \mathcal{L}^{-1} \{ G(s) \} </math>
the following table is a list of properties of unilateral Laplace transform:<ref>{{harvnb|Korn|Korn|1967|pp=226–227}}</ref>
{| class="wikitable"
|+ Properties of the unilateral Laplace transform
!
! Time domain
! 's' domain
! Comment
|-
! [[Linearity]]
| <math> a f(t) + b g(t) \ </math>
| <math> a F(s) + b G(s) \ </math>
| Can be proved using basic rules of integration.
|-
! Frequency domain differentiation
| <math> t f(t) \ </math>
| <math> -F'(s) \ </math>
| ''F''′ is the first [[derivative]] of ''F''.
|-
! Frequency domain differentiation
| <math> t^{n} f(t) \ </math>
| <math> (-1)^{n} F^{(n)}(s) \ </math>
| More general form, ''n''th derivative of ''F''(''s'').
|-
! [[Derivative|Differentiation]]
| <math> f'(t) \ </math>
| <math> s F(s) - f(0) \ </math>
| ''f'' is assumed to be a [[differentiable function]], and its derivative is assumed to be of [[exponential type]]. This can then be obtained by [[integration by parts]]
|-
! Second [[Derivative|Differentiation]]
| <math> f''(t) \ </math>
| <math> s^2 F(s) - s f(0) - f'(0) \ </math>
| ''f'' is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to ''f''′(''t'').
|-
! General [[Derivative|Differentiation]]
| <math> f^{(n)}(t) \ </math>
| <math> s^n F(s) - \sum_{k=1}^{n} s^{k-1} f^{(n-k)}(0) \ </math>
| ''f'' is assumed to be ''n''-times differentiable, with ''n''th derivative of exponential type. Follow by [[mathematical induction]].
|-
! [[Frequency|Frequency integration]]
| <math> \frac{f(t)}{t} \ </math>
| <math> \int_s^\infty F(\sigma)\, d\sigma \ </math>
| This is deduced using the nature of frequency differentiation and conditional convergence.
|-
! [[Integral|Integration]]
| <math> \int_0^t f(\tau)\, d\tau = (u * f)(t)</math>
| <math> {1 \over s} F(s) </math>
| ''u''(''t'') is the [[Heaviside step function]]. Note (''u'' ∗ ''f'')(''t'') is the [[convolution]] of ''u''(''t'') and ''f''(''t'').
|-
! Time scaling
| <math>f(at)</math>
| <math> \frac{1}{|a|} F \left ( {s \over a} \right )</math>
|
|-
! Frequency shifting
| <math> e^{at} f(t) \ </math>
| <math> F(s - a) \ </math>
|
|-
! Time shifting
| <math> f(t - a) u(t - a) \ </math>
| <math> e^{-as} F(s) \ </math>
| ''u''(''t'') is the [[Heaviside step function]]
|-
! [[Multiplication]]
| <math>f(t)g(t)</math>
| <math> \frac{1}{2\pi i}\lim_{T\to\infty}\int_{c-iT}^{c+iT}F(\sigma)G(s-\sigma)\,d\sigma \ </math>
| the integration is done along the vertical line Re(σ) = ''c'' that lies entirely within the region of convergence of ''F''.<ref>{{harvnb|Bracewell|2000|loc=Table 14.1, p. 385}}</ref>
|-
! [[Convolution]]
| <math> (f * g)(t) = \int_0^t f(\tau)g(t-\tau)\,d\tau</math>
| <math> F(s) \cdot G(s) \ </math>
| ''f''(''t'') and ''g''(''t'') are extended by zero for ''t'' < 0 in the definition of the convolution.
|-
! [[Complex conjugation]]
| <math> f^*(t) </math>
| <math> F^*(s^*) </math>
|
|-
! [[Cross-correlation]]
| <math> f(t)\star g(t) </math>
| <math> F^*(-s^*)\cdot G(s) </math>
|
|-
! [[Periodic Function]]
| <math>f(t)</math>
| <math>{1 \over 1 - e^{-Ts}} \int_0^T e^{-st} f(t)\,dt </math>
| ''f''(''t'') is a periodic function of [[periodic function|period]] ''T'' so that ''f''(''t'') = ''f''(''t'' + ''T''), for all ''t'' ≥ 0. This is the result of the time shifting property and the [[geometric series]].
|}
== بعض الدوال ومقابلها في تحويل [[بيير لابلاس|لابلاس]] ==
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