Re: Are there applications with functions of one variable composed of at least two different transcendental functions?

From	benj <benj.fake@gmail.com>
Newsgroups	sci.physics
Subject	Re: Are there applications with functions of one variable composed of at least two different transcendental functions?
Date	2018-07-25 20:05 -0400
Organization	Aioe.org NNTP Server
Message-ID	<pjb394$urb$1@gioia.aioe.org> (permalink)
References	<pj9o7m$5kt$1@news.albasani.net> <pja272$16n3$1@gioia.aioe.org> <pja4ea$9il$1@news.albasani.net> <pja87a$1irn$1@gioia.aioe.org> <pjamkl$ck8$1@gioia.aioe.org>

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On 7/25/2018 4:30 PM, Sergio wrote:
> On 7/25/2018 11:24 AM, benj wrote:
>> On 7/25/2018 11:19 AM, IV wrote:
>>> "Sergio"  schrieb im Newsbeitrag news:pja272$16n3$1@gioia.aioe.org...
>>>>> Are there applications in physics in which a closed-form expression
>>>>> exists for a function of one variable, where the function is a
>>>>> function composed of at least two different transcendental functions?
>>>>> Could you please name here some examples of such applications?
>>>> try   e^(i*x)
>>> Clearly, but this function term contains only one transcendental
>>> function: x \mapsto e^x. x \mapsto i*x is algebraic, not transcendental.
>>>
>>
>>
>> Unfortunately Sergio did the "kensi" stumble and didn't realize that i
>> is algebraic.
>>
>> So go with e^sinx . Duh.
>>
>> What is the real question here anyway? I seem to recall this same
>> discussion sometime in the past on USENET. Was that you?
>>
>>
> 
> I threw a curve ball to see what he would do, open the door for ideas,
> etc, x could be complex, or Pi, or Pi^(x*i)
> 
>    anyhow I dug this up from wiki -
> 
> Transcendental number
>  From Wikipedia, the free encyclopedia
> 
> In mathematics, a transcendental number is a real or complex number that
> is not algebraic—that is, it is not a root of a nonzero polynomial
> equation with integer (or, equivalently, rational) coefficients. The
> best-known transcendental numbers are π and e. Though only a few classes
> of transcendental numbers are known (in part because it can be extremely
> difficult to show that a given number is transcendental), transcendental
> numbers are not rare. Indeed, almost all real and complex numbers are
> transcendental, since the algebraic numbers are countable while the sets
> of real and complex numbers are both uncountable. All real
> transcendental numbers are irrational, since all rational numbers are
> algebraic. The converse is not true: not all irrational numbers are
> transcendental; e.g., the square root of 2 is irrational but not a
> transcendental number, since it is a solution of the polynomial equation
> x2 − 2 = 0. Another irrational number that is not transcendental is the
> golden ratio, φ {\displaystyle \varphi } \varphi or ϕ {\displaystyle
> \phi } \phi , since it is a solution of the polynomial equation x2 − x −
> 1 = 0.
> 
Um, yes. A curve because at first glance xi is not complex and appears 
to only be imaginary. The catch of course is that yes indeed x could be 
complex.

However if xi is simply imaginary. then we find online:
-=-=-=-=
"Well, we know that i^2 = −1, so i is the solution to:

x^2 + 1 = 0

When x = i we get −1 + 1 = 0

So the imaginary number i is an Algebraic Number"
-=-=-=-=-=
However if x is a complex number it changes things.

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