Fixed point: Difference between revisions
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A '''fixed point''' is a point in the domain of a [[function (mathematics)|function]] that is mapped to itself. Thus, a fixed point of a function ''f'' is a point ''x'' such that <math>f(x) = x</math>. | A '''fixed point''' is a point in the domain of a [[function (mathematics)|function]] that is mapped to itself. Thus, a fixed point of a function ''f'' is a point ''x'' such that <math>f(x) = x</math>. | ||
==Examples== | ==Examples== | ||
The squaring function is the function that maps each number to its [[square (arithmetics)|square]], so a fixed point of the squaring function is a number that equals its square. There are two such numbers: 0 and 1. | The squaring function is the function that maps each number to its [[square (arithmetics)|square]], so a fixed point of the squaring function is a number that equals its square. There are two such numbers: 0 and 1. | ||
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in the complex <math>z</math>-plane | in the complex <math>z</math>-plane | ||
]]<!-- | ]]<!-- | ||
[[Image:FixedPointsExpe00.jpg|300px|right|thumb|Graphical search for fixed points of [[exponential]]. Function | [[Image:FixedPointsExpe00.jpg|300px|right|thumb|Graphical search for fixed points of [[exponential]]. Function <math>f=|z-\exp(z)|</math> is shown with levels | ||
<math>f=0.5</math>, | |||
<math>f=1</math>, | |||
<math>f=2</math>, | |||
<math>f=4</math>, | |||
<math>f=8</math>, | |||
<math>f=16</math> | |||
in the complex | in the complex <math>z</math>-plane | ||
]]!--> | ]]!--> | ||
[[Image:FixedPointsExpe00.jpg|300px|right|thumb|Fig.2. The same as FIg.1 for function <math>f=|z-\exp(z)|</math> | [[Image:FixedPointsExpe00.jpg|300px|right|thumb|Fig.2. The same as FIg.1 for function <math>f=|z-\exp(z)|</math> | ||
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i.e., solutions of the equation | i.e., solutions of the equation | ||
:(10) <math> L=\ln(L) </math> | :(10) <math> L=\ln(L) </math> | ||
There are no real solutions for this equation, but there are two complex- | There are no real solutions for this equation, but there are two complex-conjugated solutions | ||
<math>L_{\rm e}</math> and <math>L_{\rm e}^*</math>. However, the value of <math>L_{\rm e}</math> cannot be estimated well from the figure (1), but the | <math>L_{\rm e}</math> and <math>L_{\rm e}^*</math>. However, the value of <math>L_{\rm e}</math> cannot be estimated well from the figure (1), but the straightforward iteration allows the precise estimate. Few hundreds of iterations are sufficient to get error of order of last significant figure in the | ||
approximation | approximation | ||
:(11) <math>L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i</math> | :(11) <math>L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i</math> | ||
===Fixed points of exp are not the same as those of ln=== | ===Fixed points of exp are not the same as those of ln=== | ||
Fixed points of logarithm should not be | Fixed points of logarithm should not be confused with fixed points of exponential, shown in FIgure 2. | ||
These fixed points are solutions of equations | These fixed points are solutions of equations | ||
:(12) <math>L=\exp(L)</math> | :(12) <math>L=\exp(L)</math> | ||
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<math>b=\sqrt{2}</math>, | <math>b=\sqrt{2}</math>, | ||
<math>b=\exp(1/\rm e)</math>, and | <math>b=\exp(1/\rm e)</math>, and | ||
<math>b=2</math>. In the last case, there are no real solutions, but the complex fixed points are complex numbers <math>L_2</math> and <math>L_2^*</math>. Within few hundred iterations of equation (14), they can be | <math>b=2</math>. In the last case, there are no real solutions, but the complex fixed points are complex numbers <math>L_2</math> and <math>L_2^*</math>. Within few hundred iterations of equation (14), they can be approximated with many decimal digits; | ||
:(15)<math> L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i | :(15)<math> L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i | ||
</math>. | </math>. | ||
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(32) <math> F(z)=L+\varepsilon(z) + o \big(\varepsilon(z)^{2}\big) </math> | (32) <math> F(z)=L+\varepsilon(z) + o \big(\varepsilon(z)^{2}\big) </math> | ||
where | where | ||
(33) <math> \varepsilon(z)=\exp(Qz+r) </math> | (33) <math> \varepsilon(z)=\exp(Qz+r) </math> | ||
<math>Q</math> and <math>r</math> are fixed complex numbers, dependent on <math>b</math>. | <math>Q</math> and <math>r</math> are fixed complex numbers, dependent on <math>b</math>. | ||
Possible values of fixed points <math>L</math> are shown with thin black curves versus logarithm on base <math>b</math>. | Possible values of fixed points <math>L</math> are shown with thin black curves versus logarithm on base <math>b</math>. | ||
At <math>\ln(b)>1/\rm e</math>, the fixed points are complex; the real part is shown with solid curve, the imaginary part is shown with dashed curve. | At <math>\ln(b)>1/\rm e</math>, the fixed points are complex; the real part is shown with solid curve, the imaginary part is shown with dashed curve. | ||
Green curves in Figure 4 represent the parameter <math> Q</math> in (33), again, the | Green curves in Figure 4 represent the parameter <math> Q</math> in (33), again, the dashed curve shows the imaginary part, and | ||
real and imaginary parts of the asymptotic period <math> 2\pi{ \rm i}/Q</math>. See article [[tetration]] for details. | real and imaginary parts of the asymptotic period <math> 2\pi{ \rm i}/Q</math>. See article [[tetration]] for details. | ||
==Expression through the special functions== | |||
In some range of values of <math>b</math>, and in particular, for <math>b>1</math>, the fixed point <math>L</math> of <math>\log_b</math> can be expressed through the [[LambertW]] function, | |||
: <math>L=-\mathrm{LambertW}(-\ln(b))/\ln(b)</math> | |||
Function LamnertW is implemented in many [[programming languages]]. | |||
The special name [[Filog]] is suggested for the function, that expresses one of the fixed point of logarithm through the logarithm of its base | |||
<ref name="filog"> | |||
http://tori.ils.uec.ac.jp/TORI/index.php/Filog <math>~ f=\mathrm{Filog}(z)</math> is solution of the equation <math>\log_b(f)\!=\!f </math> | |||
for base <math>b\!=\!\exp(z)</math> | |||
</ref>: | |||
: <math>L=\mathrm{Filog}(\ln(b))</math> | |||
Another fixed point can be expressed as <math>\mathrm{Filog}(\ln(b)^*)^*</math><br> | |||
Function Filog is simply related with the [[Tania function]]: | |||
: <math>\displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z} </math> | |||
This representation can be used in the whole complex plane; it is more robust than the representation above through the [[LambertW]] function. | |||
==[[Projector(mathematics)|Projector]]s== | ==[[Projector(mathematics)|Projector]]s== | ||
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Rev. Sci. Instrum. 74, 1 (2003); DOI:10.1063/1.1523642 | Rev. Sci. Instrum. 74, 1 (2003); DOI:10.1063/1.1523642 | ||
</ref>. | </ref>. | ||
Amplitude and phase in ultra-short pulses can be measured only with another ultra-short pulse; so, some nonlinear interaction is used to recover the temporal behavior of the pulse. The set of equations (usually two) is solved numerically, projecting the probe function to the spaces of solutions of each equation sequentially. | |||
==Notes== | ==Notes== | ||
{{reflist}}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 06:00, 17 August 2024
A fixed point is a point in the domain of a function that is mapped to itself. Thus, a fixed point of a function f is a point x such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x} .
Examples
The squaring function is the function that maps each number to its square, so a fixed point of the squaring function is a number that equals its square. There are two such numbers: 0 and 1.
In similar way, 0 is fixed point of sine function, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(0)=0} .
The function f can also be a linear operator. In this case, the fixed point is an eigenvector with eigenvalue equal to unity. For instance, the exponential function is a fixed point of the differential operator because the derivative of ex is again ex. Similarly, the Gaussian exponential exp(−x2/2) is a fixed point of the Fourier transform operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}} defined by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}[f](p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) \exp(-{\rm i}px) \, {\rm d}x. } [1]
Fixed points of exponential and fixed points of logarithm
Fixed points can be searched graphically. Fig.1 shows the graphical search of fixed points of logarithm, i.e., solutions of the equation
- (10) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\ln(L) }
There are no real solutions for this equation, but there are two complex-conjugated solutions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e}^*} . However, the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e}} cannot be estimated well from the figure (1), but the straightforward iteration allows the precise estimate. Few hundreds of iterations are sufficient to get error of order of last significant figure in the approximation
- (11) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i}
Fixed points of exp are not the same as those of ln
Fixed points of logarithm should not be confused with fixed points of exponential, shown in FIgure 2. These fixed points are solutions of equations
- (12) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\exp(L)}
They can be expressed also as solution of equation
- (13) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\log(L)+2\pi \!~{\rm i} ~m~} for some integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}
For example, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=1}
- (13)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e, 1}\approx 2.062277729598284 + 7.5886311784725127 \!~\rm i }
is fixed point of the exponential, but is not fixed point of natural logarithm.
In the case of exponential and natural logarithm, all fixed points are complex. However, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} the real fixed points exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1< b \le \exp(1/\rm e)} . For example, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/ \rm e)>} , number e is fixed point of both, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp_b} ; and at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , numbers 2 and 4 are their fixed points.
FIg.3. Example of graphic solution of equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\log_b(x)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} (two real solutions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4} ), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm e)} (one real solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\rm e} ) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} (no real solutions).
Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=f(x)=x} in the left hand side of equation
- (14) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=\log_b(z)}
the colored curves represent the function in the right hand side for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm e)} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} . In the last case, there are no real solutions, but the complex fixed points are complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2^*} . Within few hundred iterations of equation (14), they can be approximated with many decimal digits;
- (15)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i } .
Such evaluation can be performed in real time with any high level algorithmic language – Maple, Mathematica, that allow to specify number of correct decimal digits desirable in the result.
The mathematical proof of existence of the fixed points of the exponential can be found in the old paper by Helmuth Kneser [2]. Recently, Henryk Trappmann had generalized this proof for the exponentiation of arbitrary real base, but his theorem is not yet uploaded.
Tetration
The fixed points of logarithm determine the asymptotic properties of analytic extension of tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_b} . In some range of the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -plane, the tetration can be approximated with asymptotic
(32) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z)=L+\varepsilon(z) + o \big(\varepsilon(z)^{2}\big) }
where
(33) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon(z)=\exp(Qz+r) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} are fixed complex numbers, dependent on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . Possible values of fixed points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} are shown with thin black curves versus logarithm on base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(b)>1/\rm e} , the fixed points are complex; the real part is shown with solid curve, the imaginary part is shown with dashed curve.
Green curves in Figure 4 represent the parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} in (33), again, the dashed curve shows the imaginary part, and real and imaginary parts of the asymptotic period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi{ \rm i}/Q} . See article tetration for details.
Expression through the special functions
In some range of values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , and in particular, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1} , the fixed point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} can be expressed through the LambertW function,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=-\mathrm{LambertW}(-\ln(b))/\ln(b)}
Function LamnertW is implemented in many programming languages.
The special name Filog is suggested for the function, that expresses one of the fixed point of logarithm through the logarithm of its base [3]:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\mathrm{Filog}(\ln(b))}
Another fixed point can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Filog}(\ln(b)^*)^*}
Function Filog is simply related with the Tania function:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \mathrm{Filog}(z)= \frac{\mathrm{Tania}\!\big(\ln(z)-1-\mathrm{i}\big)}{-z} }
This representation can be used in the whole complex plane; it is more robust than the representation above through the LambertW function.
Projectors
Functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} on the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} is called Projector, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^2 \phi=P \phi} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \in \Psi} .
Any function of subset of functions is fixed point of projector .
In common use are projectors of the 3-dimensional space to 2-dimensional space. Such projectors allow to make flat pictures of 3-dimensional objects. Any point at the plane of image is fixed point of such a projector.
Common fixed points
The same element may be fixed point of several functors.
Sometimes, such common fixed points can be approximated using iterations. If there are two projectors and , then the sequence of elements, approximating the common fixed point of these projectors can be constructed in the following way:
Such a sequence is used to approximate the amplitude-phase behavior of ultra-short pulses in the FROG systems [4][5]. Amplitude and phase in ultra-short pulses can be measured only with another ultra-short pulse; so, some nonlinear interaction is used to recover the temporal behavior of the pulse. The set of equations (usually two) is solved numerically, projecting the probe function to the spaces of solutions of each equation sequentially.
Notes
- ↑ The expression on the left-hand side means the value at p of the Fourier transform ; the Fourier transform is the result of applying the operator to the function f.
- ↑ Helmuth Kneser. "Reelle analytische Lösungen der Gleichung und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik 187: 8-67.
- ↑ http://tori.ils.uec.ac.jp/TORI/index.php/Filog is solution of the equation for base
- ↑ Rick Trebino, Kenneth W. DeLong, David N. Fittinghoff, John N. Sweetser, Marco A. Krumbügel, and Bruce A. Richman. Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating. Rev. Sci. Instrum. 68, 3277 (1997); DOI:10.1063/1.1148286
- ↑ Giulio Cerullo and Sandro De Silvestri. Ultrafast optical parametric amplifiers. Rev. Sci. Instrum. 74, 1 (2003); DOI:10.1063/1.1523642
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/017a97e3f0c147c2decb79f1d78e029f91ef95b3)
![{\displaystyle {\mathcal {F}}[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d84a4ecfeb48a40b6273d5b9e11c1bbe685455d8)
