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A possible more general form of Euler's identity

  1. Oct 21, 2012 #1
    Hi, I cannot find any other reference to this formula:

    sin(x)/cos(x-1)

    it seems to fit with Euler's identity as given by Wikipedia. Euler's identity is a special case of this identity equation.

    I've actually posted this on the Wikipedia page to see if I can get confirmation of this or have someone tear it down.

    It also has some bizarre properties between -∏-3 ≤ x ≤ ∏-3.

    More information on this function from wolframalpha.com

    My maths is very rusty and originally derived the formula as I thought it would make a pretty graph which I think it does. Only since reading up on prime numbers did I revisit this and check for values less than 1.

    I'm startled at how simple looking the function is, yet unable to find it in this form anywhere.

    I'd be interested to hear any opinions on it and will do my best to answer any questions about it.

    Thanks!
     
    Last edited: Oct 21, 2012
  2. jcsd
  3. Oct 21, 2012 #2
    What do you mean? How does something "fit" with Euler's Identity, and what do you mean that Euler's Identity is a special case of [itex]\sin x / \cos x^{-1}[/itex]?
     
  4. Oct 21, 2012 #3

    pwsnafu

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    You should be posting that on the Talk page, not in the article itself. I would suggest moving it ASAP before a moderator deletes and locks the page.
     
  5. Oct 22, 2012 #4

    micromass

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    What equation?? You just posted the function [itex]\frac{\sin(x)}{\cos(1/x)}[/itex]. How is this an equation?
     
  6. Oct 22, 2012 #5
    OK, if you put y= in front of the function it becomes an equation...

    A function maps an input to an output, the line of a graph maps a series of inputs to a series of outputs where the points on the line express an equality between the input and output of the function plotted.
     
  7. Oct 22, 2012 #6
    You are right, taken down now and should not have put there in the 1st place!
     
  8. Oct 22, 2012 #7
    I probably should have considered this for longer before posting this somewhere such as here. I did write an explanation of why I thought this may be a more general case for Euler's identity, but my maths has not been tested for some time and I cannot coherently communicate my arguments which are heavily guided by intuition at this stage.

    I have not proved it or even attempted proof, yet, but posted this here to see if anyone else had thoughts about this function or may be interested in investigating some of it's properties for themselves.

    There are a lot of strange things about this which I want to spend more time pouring over and looking into some formal proofs before I go any further with this.

    There's also the distinct possibility that I'm completely wrong, but am still intrigued by the details of this expression.

    Unfortunately I'm up to my eyeballs in other more mundane work which has to take priority over this at the moment.

    Looking at the information provided by wolframalpha.com there is an equivalent way of writing this function in terms of e and i without any trig functions. This will most likely be my starting point to see how it can be rearranged and if there are particular values for x that could give Euler's identity.

    Also the binomial(?) expansion of the function has Mersenne numbers as denominators and "fits" (or matches!) this arithmetic progression: http://oeis.org/A036282 also linked to Reinmann Zeta functions. I don't grasp exactly what these are yet, but understand they are of significance to the prime numbers which could be construed the building blocks of reality!

    As I stated it was thinking about prime numbers that led me here.

    I have a hunch that the breaks in the plot to +/- infinty for -∏-3 ≤ x ≤ ∏-3 which approach a period of 0 towards x=0 are linked to the primes.

    I also like the relationship between order and chaos that this seems to give. With chaos approaching x=0 and the order of the maximums of the wave function on (or near?) x=+/-y.

    I will post my findings here when I get some time, but back to the grindstone for me now.
     
  9. Oct 22, 2012 #8
    I meant to put a question mark on the end of the title of this thread too, but was rather tired when I did it.
     
  10. Oct 22, 2012 #9
    Oh, and a nicer online graph plot of the function/equation:

    cos(x)/sin(1/x)

    or also written as:

    -(i (e^(-i x)+e^(i x)))/(e^(-i/x)-e^(i/x))

    see: http://fooplot.com/plot/5afahic6zv for zoom-in-able visualisation

    there are some interference pattern effects to be wary of with the digital imaging of this though picking sensible scales for the axes helps...

    - bizarre for -1>x>1
     
  11. Oct 22, 2012 #10
    It can't be a more general case of Euler's identity because it's not an identity. Euler's identity says that something is equal to something else; you just wrote an expression.

    More likely, they're just the result of the denominator going to zero.

    Your function is not a dynamical system; it is not chaotic.
     
  12. Oct 22, 2012 #11
    Thanks, excellent points and well made!

    What about this equation then:

    cos(x)/sin(1/x)=sin(y)/cos(1/y)
     
  13. Oct 22, 2012 #12
    Getting a little bit complicated now, but plot here: http://wolfr.am/VkwFv8 is truly bizarre?

    I now need better software on my computer as would like more detailed plot and the simple online tool will only plot in terms of y= ....

    I don't fancy finding y from that equation! I must get back to my work too...
     
  14. Oct 22, 2012 #13
    For which values of x and y? And what does it have to do with Euler's identity?
     
  15. Oct 22, 2012 #14
    All of them. Don't know about Euler's identity, but this looks a bit like the fingerprint of the universe:

    https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-ash4/400225_10151029543626116_786265305_n.jpg

    I have never seen anything this strange come from a relatively simple trigaonametric function. These are plots of real values, it will be interesting to see what happens when this is applied to complex numbers.
     
  16. Oct 22, 2012 #15
    No, not for every possible value of (x, y). Not for x = 5 and y = 3.

    Uh, why? What is that supposed to mean, exactly?

    It's not a function, it's a relation. You've found a relation with a weird looking graph; there are lots of them.
    It's only interesting if it means something, and you haven't demonstrated that this expression of yours means or does anything.
     
  17. Oct 22, 2012 #16
    oh right, never seen one. that knocks that on the head then.
     
  18. Oct 22, 2012 #17
    they're very pretty though!

    Selection_048.png
     
  19. Oct 22, 2012 #18
    https://fbcdn-sphotos-e-a.akamaihd.net/hphotos-ak-snc6/253026_10151029575451116_1780170426_n.jpg
     
  20. Oct 22, 2012 #19
    i wander what they all look like superimposed on each other.

    How many are there?

    cos(x)/cos(1/y)=sin(1/x)/sin(y)
    cos(x)/cos(1/y)=sin(y)/sin(1/x)
    cos(x)/sin(1/x)=sin(y)/cos(1/y)
    cos(x)/sin(1/x)=cos(1/y)/sin(y)
    cos(x)/sin(y)=cos(1/y)/sin(1/x)
    cos(x)/sin(y)=sin(1/x)/cos(1/y)
    sin(y)/cos(x)=cos(1/y)/sin(1/x)
    sin(y)/cos(x)=sin(1/x)/cos(1/y)

    i make that 8
     
  21. Oct 22, 2012 #20
    You're just taking combinations of sines and cosines. There's nothing special or unusual about this.
     
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