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''' <math> L </math> of a [[functor]] <math> F </math> is a solution of equation
==Examples==
: <math>F[L]=L</math>,
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.
that is a point that is mapped to itself.


==Simple examples==
In similar way, 0 is fixed point of [[sine]] function, because <math>\sin(0)=0</math>.


===Elementary functions===
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 e<sup>''x''</sup> is again e<sup>''x''</sup>. Similarly, the [[Gaussian exponential]] exp(−''x''<sup>2</sup>/2) is a fixed point of the [[Fourier transform]] operator <math>\mathcal{F}</math> defined by
In particular, functor can be elementaty function. For example, 0 and 1 are fixed points of function [[sqrt]], because <math>\sqrt{0}=0</math> and
:<math> \mathcal{F}[f](p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) \exp(-{\rm i}px) \, {\rm d}x. </math> &nbsp;&nbsp;&nbsp;&nbsp;<ref>The expression <math>\mathcal{F}[f](p)</math> on the left-hand side means the value at ''p'' of the Fourier transform <math>\mathcal{F}[f]</math>; the Fourier transform is the result of applying the operator <math>\mathcal{F}</math> to the function ''f''.</ref>
<math>\sqrt{1}=1</math>.
 
In similar way, 0 is fixed point of [[sine]] function, because <math>\sin(0)=0</math> .
 
===Operators===
The functor in the equation, defining a fixed point, can be a [[linear operator]]. In this case, the fixed point of the functor <math>F</math> is its [[eigenfunction]] with [[eigenvalue]] equal to unity.  
 
[[Exponential]] if fixed point or [[operator of differentiation]] D,
because
<math>{\rm D}~ \exp(x) =\exp'(x)=\exp(x)</math>
 
The [[Gaussian exponential]]
:<math>L(x)=\exp(-x^2/2)</math> , with <math>x \in \mathbb{R}</math> and <math>\mathbb{R}</math> the [[real number|reals]],
is fixed point of the [[Fourier operator]], defined with its action on a function <math>g</math>:
:(3) <math>F(g)(p)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}
g(x)\exp(-{\rm i}px) {\rm d}x </math>
in general, functors have no need to be linear, and there is no [[associativity]]
at application of several functors in row, and parenthesis are necessary in the left hand side of eapression (3).
<ref name="ambiguity">
Note that that there is certain [[ambiguity]] in commonly used writing of mathematical formulas, omiting sign * of multiplication; in equation (3), expression  
<math>F(g)(p)</math> does not mean that <math>F(g)</math> is multiplied to value of <math>p</math>; it means that result <math>F(g)</math> of action of operator <math>F </math> on function <math>g</math>, which is also a function, is [[evaluate]]d at argument <math>p</math>.
</ref>


==Fixed points of [[exponential]] and fixed points of [[logarithm]]==
==Fixed points of [[exponential]] and fixed points of [[logarithm]]==
Line 42: Line 20:
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 <math>f=|z-\exp(z)|</math> is shown with levels  
[[Image:FixedPointsExpe00.jpg|300px|right|thumb|Graphical search for fixed points of [[exponential]]. Function &lt;math>f=|z-\exp(z)|&lt;/math> is shown with levels  
<math>f=0.5</math>,
&lt;math>f=0.5&lt;/math>,
<math>f=1</math>,
&lt;math>f=1&lt;/math>,
<math>f=2</math>,
&lt;math>f=2&lt;/math>,
<math>f=4</math>,
&lt;math>f=4&lt;/math>,
<math>f=8</math>,
&lt;math>f=8&lt;/math>,
<math>f=16</math>
&lt;math>f=16&lt;/math>
in the complex <math>z</math>-plane
in the complex &lt;math>z&lt;/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>
Line 57: Line 35:
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-congjugated solutions  
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 straigtgorward iteration allows the precise estimate. Few hundreds of iterations are sufficient to get error of order of last significant figure in 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 confised with fixed points of exponential, shown in FIgure 2.
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>
Line 79: Line 57:
2 and 4 are their fixed points.  
2 and 4 are their fixed points.  


[[Image:ExampleEquationLog01.png|300px|right|thumb|FIg.3. Example of graphic solution of equation  
{{Image|ExampleEquationLog01.png|right|300px|FIg.3. Example of graphic solution of equation  
<math>x=\log_b(x)</math> for
<math>x=\log_b(x)</math> for
<math>b=\sqrt{2}</math> (two real solutions, <math>x=2</math> and <math>x=4</math>),
<math>b=\sqrt{2}</math> (two real solutions, <math>x=2</math> and <math>x=4</math>),
<math>b=\exp(1/\rm e)</math> (one real solution <math>x=\rm e</math>)
<math>b=\exp(1/\rm e)</math> (one real solution <math>x=\rm e</math>)
<math>b=2</math> (no real solutions).]]
<math>b=2</math> (no real solutions).}}
Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function <math>f=f(x)=x</math> in the left hand side of equation
Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function <math>f=f(x)=x</math> in the left hand side of equation
:(14) <math> z=\log_b(z)</math>
:(14) <math> z=\log_b(z)</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 approcimated with many decimal digits;   
<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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| volume=187
| volume=187
| pages=8-67
| pages=8-67
}}</ref>. Recently, [[Genryk Trappmann]] had generalized this proof for the exponentiation of arbitrary real base, but his theorem is not yet uploaded.
}}</ref>. Recently, [[Henryk Trappmann]] had generalized this proof for the exponentiation of arbitrary real base, but his theorem is not yet uploaded.


===Tetration===
===Tetration===
[[Image:TetrationAsymptoticParameters01.jpg|700px|left|thumb|FIg.4. parameters of asymptotic of tetration versus logarithm of the base]]
{{Image|TetrationAsymptoticParameters01.jpg|left|700px|FIg.4. parameters of asymptotic of tetration versus logarithm of the base}}
The fixed points of logarithm determine the asymptotic properties of analytic extension of [[tetration]] <math>F_b</math>. In some range of the
The fixed points of logarithm determine the asymptotic properties of analytic extension of [[tetration]] <math>F_b</math>. In some range of the
complex <math>x</math>-plane, the tetration can be approximated with asymptotic
complex <math>x</math>-plane, the tetration can be approximated with asymptotic
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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 dased curve shows the imaginary part, and
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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===Common fixed points===
===Common fixed points===
The same element <math>f</math> may be fixed point of several projectors.
The same element <math>f</math> may be fixed point of several functors.


Sometimes, such common fixed points can be approximated using iterations.
Sometimes, such common fixed points can be approximated using iterations.
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: <math>f_n=P_1\Big(P_2(f_{n-1})\Big) </math>
: <math>f_n=P_1\Big(P_2(f_{n-1})\Big) </math>
Such a sequence is used to approximate the [[amplitude-phase behavior]] of [[ultra-short pulses]] in the
Such a sequence is used to approximate the [[amplitude-phase behavior]] of [[ultra-short pulses]] in the
[[Frequency Resolved Optical Gating|FROG]] systems <ref name="frog">
[[Frequency Resolved Optical Gating|FROG]] systems <ref name="frog">  
Amplitide 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.</ref>.
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
</ref><ref>
Giulio Cerullo and Sandro De Silvestri.
Ultrafast optical parametric amplifiers.
Rev. Sci. Instrum. 74, 1 (2003); DOI:10.1063/1.1523642
</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==
<references/>
{{reflist}}[[Category:Suggestion Bot Tag]]
 
==See also==
*[[exponential]]
*[[Linear operators]]
*[[Eigenfunciton]]
*[[eigenvalue]]
*[[tetration]]
*[[projector]]
 
[[Category:Mathematics]]

Latest revision as of 06:00, 17 August 2024

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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

FIg.1. 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-\ln(z)|} is shown with levels 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=0.5} , 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=1} , 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=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 f=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 f=8} , 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=16} in 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 z} -plane
Fig.2. The same as FIg.1 for 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=|z-\exp(z)|}

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.

© Image: Dmitrii Kouznetsov
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

FIg.4. parameters of asymptotic of tetration versus logarithm of the base

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 . Possible values of fixed points are shown with thin black curves versus logarithm on base . At , 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 in (33), again, the dashed curve shows the imaginary part, and real and imaginary parts of the asymptotic period . See article tetration for details.

Expression through the special functions

In some range of values of , and in particular, for , the fixed point of can be expressed through the LambertW function,

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]:

Another fixed point can be expressed as
Function Filog is simply related with the Tania function:

This representation can be used in the whole complex plane; it is more robust than the representation above through the LambertW function.

Projectors

Functor on the space is called Projector, if for all .

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

  1. ↑ 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.
  2. ↑ Helmuth Kneser. "Reelle analytische Lösungen der Gleichung und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik 187: 8-67.
  3. ↑ http://tori.ils.uec.ac.jp/TORI/index.php/Filog is solution of the equation for base
  4. ↑ 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
  5. ↑ Giulio Cerullo and Sandro De Silvestri. Ultrafast optical parametric amplifiers. Rev. Sci. Instrum. 74, 1 (2003); DOI:10.1063/1.1523642
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