I found a very elegant proof of a very well known calculus level result. The result, I don't want to say which yet, has been definitely proven by other means. I researched all the proofs, in a few books and all google, never have I seen my result there. I asked a math professor about it, but he was only aware of one famous proof, I didn't tell him mine though. I don't want to sound like this is big, but it's just a small result that could fit on a page. The proof attacks the problem from a completely different angle. So what is typically done in a situation like this. Would it be a good idea to write a calculus level paper? If so can you recommend a journal. All journals I've seen are in differential topology and up. Appreciate any info.
You got to be careful about going overboard on something like that. I have know students in that situation who would not trust a professor to honestly handle this. But, I don't think people are really out to steal these kind of things, so I suggest try and find someone in authority you can trust to take a look at it.
Well, waht, typically people are no so much afraid to explain a little bit about their finding. If this is a known result, you could at least tell which one it is. With some information we might give you some concrete suggestion. Consider for example the American Journal of Physics (teachers): you can find there many derivations of old results, sometimes elegant or sometimes insightful. A similar journal for maths could be ok. But it al depends on the topic. It may also be that your new approach is the deciding element, for example when this approach could have other applications. So, waht, don't frustate people by saying without saying. You have to decide if you speak or not.
don't bother ... to be honest, nobody will care about it, as the result has been proven before ... it'll be received with a slight nod and a "yes of course" in the best-case-scenario ... if it's not for the spotlight, and you want to publish your thoughts, then go for it ... but if you are doing it for acclaim then do not bother, as the modern standards of rigor would require your paper to be much longer and clearer than it already is ...
I doubt it's something that would get published in a top journal (although, I'm not familiar with mathematical journals as such). Of course if you want to write it for experience, then go ahead and then either show it to your professor for possible feedback, or put it on the arxiv, or both.
It is highly unlikely that a maths journal will be remotely intertested in this unless the method has great potential. However, there are plenty of mathematical magazines and periodicals that might be interested, though I can't name them off the top of my head. I;m sure the AMS, or AMA have opportunities to publish such things in their magazines.
Thanks for the feedback. I checked out arxiv and AMS and frankly it would be a joke to sent it there in my opinion, as the level of sophistication required to publish is light years ahead of my little proof. So here it is. I've attached a rough draft, without any references. It is basically a new proof of Euler's Identity. Feel free to review it.
On your own, you have some merit; you defined the logarithmic function for complex numbers by means of its definition as a real limit. However, this is not novel. The Taylor expansion of the logarithmic function has the same limit, real or complex, (as we add more terms) as the fraction [tex]\frac{a^{h} - 1}{h}[/tex] as h goes to 0. This means that taking the definition from either the expansion of ln or its definition as a limit is all the same; this is insinuated in the use of the Taylor expansion of ln to define the logarithm of a complex number.
The formula for log actually came from a very weird calculus book that vagues touches it. (that's why I said no references) My idea is to use the formula to take the log of cos(x)+i sin(x). What is also interesting about the log function, if you calculate the inverse of log, you are going to get the standard compound interest formula for "e". The Taylor series for ln(x+1) converges for only (-1 < x <= 1). Where as this limit formula converges for x > 0. I'm not sure how to prove it converges for complex numbers, but it seems to. Thanks. Thanks.
Waht, I must say this is not really a new and elegant demonstration. It serves essentially as an exercice to link different point of view. Personally, I even consider that this formula does not really need a demonstration, in the sense that the group-properties of the complex numbers are the only really important think. The fact that these properties can be linked to group properties of the (plane) rotations and therefore to some trigonometry is essentially a manifestations that groups can be represented in different ways. But the most important think is that you had fun with that and that it gave you some confidence in yourself and in the consistency of mathematics. I suggest you to go forward. Have a try at the exponential of matrices. Have more fun from the fact that matrices don't necessarily commute for multiplication.
In your proof you used the real variable version of de Moivre's identity, which uses Euler's identity in its proof (at least, in the proof I'm aware of). So unless you can provide an alternative proof of de Moivre, your current proof is circular I'm afraid.
Abraham de Moive discovered his formula without using Euler's identity, and before Euler's identity. If I recall correctly, he used series expansions.
I was under the impression that the original version of de Moivre's identity was stated for (nonnegative) integers n only, and was proved by induction. The general version came later. Feel free to correct me if I'm wrong.
Look here, fourth expansion: http://www.math.com/tables/expansion/log.htm I do not doubt that some proofs use this expansion as a means to define the logarithm of a complex number.