Rosso: /* Causality and Kramers-Kronig relations */

2021-11-10T15:47:59Z

Causality and Kramers-Kronig relations

← Older revision		Revision as of 17:47, 10 November 2021
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	=== Causality and Kramers-Kronig relations ===		=== Causality and Kramers-Kronig relations ===
	'''Q12:''' Using the expression of the Fourier transform of a convolution and the sign function defined above, show the following property of Fourier transform:		'''Q12:''' Using the expression of the Fourier transform of a convolution and the sign function defined above, show the following property of Fourier transform:
	<center> If <math>g(t)=0</math> <math>\forall t<0</math>, then <math>\quad\hat g(\omega)=\frac{1}{i\pi}\,{\rm PV} \int_\mathbb{R}d\xi \,\frac{~~\operatorname{Im}~~ \hat g(\xi)}{\xi -\omega}</math>.		<center> If <math>g(t)=0</math> <math>\forall t<0</math>, then <math>\quad\hat g(\omega)=\frac{1}{i\pi}\,{\rm PV} \int_\mathbb{R}d\xi \,\frac{ \hat g(\xi)}{\xi -\omega}</math>.
	</center>		</center>
	By separating the real and imaginary part of this equation, we obtain Kramers-Kronig relations:		By separating the real and imaginary part of this equation, we obtain Kramers-Kronig relations:

Rosso at 14:01, 9 November 2021

2021-11-09T14:01:54Z

Rosso at 17:33, 13 October 2021

2021-10-13T17:33:31Z

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← Older revision		Revision as of 16:01, 9 November 2021
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	x(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d t' \, g(t-t')f(t'),		x(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}d t' \, g(t-t')f(t'),
	</math></center>		</math></center>
	where <math>g(t)</math> is the function having <math>\hat g(\omega)</math> as Fourier transform. The above equation is nothing but the convolution of <math>f</math> with <math>g</math>: <math>x(t)=(f\ast g)(t)</math>.		where <math>g(t)</math> is the function having <math>\hat g(\omega)</math> as Fourier transform. The above equation is nothing but the convolution of <math>f</math> with <math>g</math>: <math>\sqrt{2\pi}x(t)=(f\ast g)(t)</math>.

	=== Holomorphism of the response and consequences ===		=== Holomorphism of the response and consequences ===
	'''Q4:''' Show that <math>g(t)</math> is the response of the system to an impulsive perturbation <math>f(t)=\delta(t)</math>.		'''Q4:''' Show that <math>g(t)</math> is the response of the system to an impulsive perturbation <math>f(t)=\sqrt{2\pi}\delta(t)</math>.
	What is the effect of a perturbation applied at time <math>s</math> on the solution at time <math>t>s</math>?		What is the effect of a perturbation applied at time <math>s</math> on the solution at time <math>t<s</math>?

	'''Q5:''' We have defined <math>\hat g (\omega)</math> for <math>\omega\in\mathbb{R}</math>, i.e. on the real axis. Show that its extension to the complex plane is holomorphic everywhere (except at the poles). Using the explicit expression for <math>\widehat g (\omega)</math> derived above, compute explicitly <math>g(t)</math>. You will have to treat separately the case of positive time and the case of negative time. Use the residue theorem.		'''Q5:''' We have defined <math>\hat g (\omega)</math> for <math>\omega\in\mathbb{R}</math>, i.e. on the real axis. Show that its extension to the complex plane is holomorphic everywhere (except at the poles). Using the explicit expression for <math>\widehat g (\omega)</math> derived above, compute explicitly <math>g(t)</math>. You will have to treat separately the case of positive time and the case of negative time. Use the residue theorem.
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	* if <math>g\in \mathbb{R} </math> is an odd function, then <math> \hat g\in i\mathbb{R} </math> is also an odd function;		* if <math>g\in \mathbb{R} </math> is an odd function, then <math> \hat g\in i\mathbb{R} </math> is also an odd function;
	* if <math> g\in i\mathbb{R}</math> is an odd function, then <math>\hat g\in \mathbb{R}</math> is also an odd function;		* if <math> g\in i\mathbb{R}</math> is an odd function, then <math>\hat g\in \mathbb{R}</math> is also an odd function;
	* if <math> g\in \mathbb{R} </math> is an even function, then <math> \hat g\in i\mathbb{R} </math> is also an even function;		* if <math> g\in \mathbb{R} </math> is an even function, then <math> \hat g\in \mathbb{R} </math> is also an even function;
	* if <math>g\in i\mathbb{R} </math> is an even function, then <math> \hat g\in \mathbb{R} </math> is also an even function.		* if <math>g\in i\mathbb{R} </math> is an even function, then <math> \hat g\in i\mathbb{R} </math> is also an even function.

	'''Q8:''' Conclude that, if <math>g</math> is real, its Fourier transform is written as <math>\hat g=\hat g ' +i\hat g ''</math>, with <math>\hat g '</math> ~~odd~~, and <math>\hat g '' </math> ~~even~~.		'''Q8:''' Conclude that, if <math>g</math> is real, its Fourier transform is written as <math>\hat g=\hat g ' +i\hat g ''</math>, with <math>\hat g '</math> even, and <math>\hat g '' </math> odd.

	'''Q9:''' Derive the same properties for the case of the oscillator.		'''Q9:''' Derive the same properties for the case of the oscillator.

T-I-2 - Revision history

Rosso: /* Causality and Kramers-Kronig relations */

Rosso at 14:01, 9 November 2021

Rosso at 17:33, 13 October 2021

Rosso: Created page with "http://lptms.u-psud.fr/wiki-cours/index.php/T-I-4"