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Intresting functions not ?
#1
I was thinking about some " intresting " functions.

Not sure how to define " intresting " though.

For example :

exp(a+bi) = a+bi

f(z) = ln( (exp(z) - z) / (z^2 + 2az + a^2 + b^2) )

Where a and b are real and z is complex.

It seems f(z) must be entire since exp(z) has only 2 fixpoints.

Another example is

f(z) = ln( (2sinh(z) - z) / z )

Forgive me for once again writing 2sinh(z) Wink

Im a 2sinholic Smile

We can get Taylor series for these kind of functions.

But I wonder if these have been studied before ?

They appear both "elementary and exotic " to me.

regards

tommy1729
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#2
Since f(z) = ln( (2sinh(z) - z) / z ) is entire we know there must be a solution to (2sinh(z) - z) / z = (1 + a_1 z)(1 + a_2 z^2)(1+a_3 z^3)... where the a_n are (complex) constants.
(Notice f(0) = 1)

It might then be intresting to consider the size and signs of the a_n.

This is a " Gottfried type " of idea I must say.

Although these products are clearly from the time mathematicians considered the so called q-series and are still investigated by some today (like me) I would like to give a link to Gottried's paper.

Im not so knowledgeable about Witt vectors , but I think this paper explaines some things. ( So that I do not need to repeat the Obvious )

I would also like to say these products have an intresting combinatorical interpretation that can be seen as a competative way against the circle method ( powers of a Taylor series ).

http://go.helms-net.de/math/musings/drea...quence.pdf

Here the importance of the ln and both the partition and divisor function occurence is well illustrated.

The " fake zero's " a_n^(1/n) are both fascinating and puzzling.

In Gottfried's paper it means the " q-product form " of exp(z) has a natural boundary smaller or equal to the unit radius BECAUSE OF EITHER THE LN OR THE A_N ( depending on your viewpoint ).

But other cases of " q-product forms " apart from exp(z) might be more complex.

Since the focus here is on functions that grow about exponential rate and also the focus on fixpoints and expansions this seems like a natural question to me.

Regards

tommy1729
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#3
(09/23/2013, 11:18 PM)tommy1729 Wrote: Not sure how to define " intresting " though.

...

exp(a+bi) = a+bi
tommy1729

I don't know what makes a function interesting either or to what extent any of these functions have been studied. However, I found the answer to exp(a+bi) = a+bi, albeit not properly at all since I used a calculator to graph a certain part ((b/sin(b))-exp(b/tan(b)) = 0) to solve for the b portion. I feel like that one must have been done before since a calculator can be used to find a and b. I could be very wrong on this.
Since there is an answer in the set of complex numbers it doesn't seem as interesting to me as it would if another set had to be used.
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#4
(03/04/2014, 02:33 PM)razrushil Wrote:
(09/23/2013, 11:18 PM)tommy1729 Wrote: Not sure how to define " intresting " though.

...

exp(a+bi) = a+bi
tommy1729

I don't know what makes a function interesting either or to what extent any of these functions have been studied. However, I found the answer to exp(a+bi) = a+bi, albeit not properly at all since I used a calculator to graph a certain part ((b/sin(b))-exp(b/tan(b)) = 0) to solve for the b portion. I feel like that one must have been done before since a calculator can be used to find a and b. I could be very wrong on this.
Since there is an answer in the set of complex numbers it doesn't seem as interesting to me as it would if another set had to be used.
Surely this has been done may times; I think I've seen this for instance in Corless et al. profound article on the LambertW-function; but also D. Shell / (?) Thron (after who the region of convergence of the infinite powertower in the comple numbers has been named) should have needed the solution for this. For me it was in the beginning of my encounter with tetration interesting to find complex fixpoints for real bases (of a continuous interval), so I made this picture (it seems to be a bit overcomplicated, but well...)
http://go.helms-net.de/math/tetdocs/real...uction.png
and
http://go.helms-net.de/math/tetdocs/real...tion_2.png


I don't understand your last remark: "if another set would be used" - what do you mean?

Gottfried



Attached Files Image(s)
   
Gottfried Helms, Kassel
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#5
I kind of forgot about the Lambert W function (never really encountered it properly, but did see it once when looking at things for tetration). Just today I noticed when trying to find what i^^n converged to as n→infinity that I ended up with x = i^x, which would fit this situation almost exactly.
As far as my last remark, I could have worded things improperly, but I meant that I would be more interested by a problem that could be solved, but not by complex numbers alone. For now it is kind of difficult to think of such problems, but I'll definitely need more math before being able to effectively work with such problems.
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