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Factorizing taylor polynomials of infinite degree

  1. Apr 3, 2009 #1
    an idea i had:


    factorizing taylor polynomials
    Can any taylor polynomial be factorized into an infinite product representation?
    I think so.

    I was able to do this(kinda) with sin(x), i did it this way.

    because sin(0)=0, there must be an x in the factorization.
    because every x of sin(x)=0 is n*pi where n is any number, there is (x+pi)(x-pi)(x+2pi)(x-2pi)... in the factorization
    the (x+n*pi)(x-n*pi) can be simplified to (x^2-n^2*pi^2)

    i know therefore that the factorized polynomial of sin(x) is in the form of

    ax(x^2-pi^2)(x^2-4pi^2)(x^2-9pi^2)(x^2-16pi^2)(x^2-25pi^2)...

    heres where i got messed up:

    a= the coefficient of the highest exponent in the series, which is n*2+1 where n is the number of factors.
    a=1/(2n+1)!

    which comes from the infinite sum representation. but as it turns out this is incorrect.(which after reasoning through it i realize that this the factorization is totally different from the infinite sum representation. and therefore i was wrong) but with help from my graphing calculator i was able use guess and check to decide the value of a.

    also, it turns out that if n even, the factorization is sinx, while if n is odd, the factorization is -sinx. this is strange.

    heres one factorization i came up with. but its only accurate between -pi and pi. graph it and see what i mean.

    sinx=(1/500000000000)x(x^2-pi^2)(x^2-4pi^2)(x^2-9pi^2)(x^2-16pi^2)(x^2-25pi^2)(x^2-36pi^2)

    it seems as though regardless of the number of factors, the polynomial is only accurate between -pi and pi.


    with the moderate success of factorizing sin(x), i decided to try it with e^x
    e^x=x^0/0!+x^1/1!+x^2/2!+x^3/3!+...

    however, the only solution of e^x=0 is negative infinity; there are not even any complex roots that i can work with.
    so does this mean that e^x non-factorizable? i cant imagine why.
     
  2. jcsd
  3. Apr 3, 2009 #2
    http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem" [Broken]
     
    Last edited by a moderator: May 4, 2017
  4. Apr 10, 2009 #3
    foxjwill's right on the money. You also have to be very careful about what you mean for an infinite product to converge. A lot of things you think should converge as infinite products dont.
     
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