Fixed point: Difference between revisions

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imported>Dmitrii Kouznetsov
imported>Dmitrii Kouznetsov
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They can be expressed also as solution of equation
They can be expressed also as solution of equation
:(13) <math>L=\log(L)+2\pi \!~\rm i ~m~</math> for some integer <math>m</math>
:(13) <math>L=\log(L)+2\pi \!~\rm i ~m~</math> for some integer <math>m</math>
For example,  
For example, at <math>m=1</math>
:(13)<math>L_{\rm e, 2}\approx  
:(13)<math>L_{\rm e, 1}\approx  
2.062277729598284 + 7.5886311784725127 \!~\rm i  
2.062277729598284 + 7.5886311784725127 \!~\rm i  
</math>  
</math>  
is fixed point of the exponential, but is not the fixed point of natural logarithm.
is fixed point of the exponential, but is not the fixed point of natural logarithm.
In the case of exponential and natural logarithm, all fixed points are complex.
However, <math>\log_b</math> the real fixed points exist at <math>1< b \le \exp(1/\rm e)</math>.
For example, at <math>b=\exp(1/ \rm e)></math>, number e is fixed point of both,
<math>\log_b</math> and <math>\exp_b</math>; and at <math>b=\sqrt{2}</math>, numbers
2 and 4 are their fixed points.
[[Image:ExampleEquationLog01.png|300px|right|thumb|FIg.3. Example of graphic solution of equation
<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=\exp(1/\rm e)</math> (one real solution <math>x=\rm e</math>)
<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
:(14) <math> z=\log_b(z)</math>
the colored curves represent the function in the right hand side for
<math>b=\sqrt{2}</math>
<math>b=\exp(1/\rm  e</math>
<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 decima ligits; 
:(15)<math> L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i
</math>.
The fixed points of logarithm determine the asymptotic properties of analytic extension of [[tetration]].


==Notes==
==Notes==
<references/>
<references/>

Revision as of 06:17, 31 May 2008

Template:Under construction

Fixed point of a functor is solution of equation

(1)

Simple examples

Elementary functions

In particular, functor can be elementaty function. For example, 0 and 1 are fixed points of function sqrt, because and .

In similar way, 0 is fixed point of sine function, because .

Operators

Functor in the equation (1) can be a linear operator. In this case, the fixed point of functor is its eigenfunction with eigenvalue equal to unity.

Exponential if fixed point or operator of differentiation D, because

The Gaussian exponential

(2) , reals

is fixed point of the Fourier operator, defined with its action on a function :

(3)

in general, functors have no need to be linear, so, there is no associativity at application of several functors in row, and parenthesis are necessary in the left hand side of eapression (3). [1]

Fixed points of exponential and fixed points of logarithm

FIg.1. is shown with levels , , , , , in the complex -plane
Fig.1. The same as FIg.1 for function

Fixed points can be searched graphically. Fig.1 shows the graphical search of fixed points of logarithm, i.e., soluitons of the equaiton

(10)

There are no real solutions fot this equation, but there are two complex-congjugated solutions and . However, the value of 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 approximation

(11)

Fixed points of logarithm should not be confised with fixed points of exponential, shown in FIgure 2. Therse fixed points are solutions of equaitons

(12)

They can be expressed also as solution of equation

(13) for some integer

For example, at

(13)

is fixed point of the exponential, but is not the fixed point of natural logarithm.

In the case of exponential and natural logarithm, all fixed points are complex. However, the real fixed points exist at . For example, at , number e is fixed point of both, and ; and at , numbers 2 and 4 are their fixed points.

FIg.3. Example of graphic solution of equation for (two real solutions, and ), (one real solution ) (no real solutions).

Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function in the left hand side of equation

(14)

the colored curves represent the function in the right hand side for . In the last case, there are no real solutions, but the complex fixed points are complex numbers and . Within few hundred iterations of equation (14), they can be approcimated with many decima ligits;

(15).

The fixed points of logarithm determine the asymptotic properties of analytic extension of tetration.

Notes

  1. Note that that there is certain ambiguity in commonly used writing of mathematical formulas, omiting sign * of multiplication; in equation (3), expression does not mean that is multiplied to value of ; it means that result of action of operator on function , whith is also function, is evaluated at argument .