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Math Help - Non-recursive formula for y^{(n)} for inverse power series

  1. #1
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    Non-recursive formula for y^{(n)} for inverse power series

    I'm interested in inverting a power series: Suppose through solving a DE for the function y(x), I ended up with a power series in y:

    \sum_{n=0}^{\infty} a_n y^n=x

    Now, we know a_n=\frac{1}{n!} \frac{d^n x}{dy^n}

    I'm just learning I can invert that series to obtain:

    y(x)=\sum_{n=0}^{\infty} b_n x^n

    with b_n=\frac{1}{n!} \frac{d^n y}{dx^n}

    But each derivative in y is related to each derivative in x by the recursive formula:

    y_n=\frac{1}{x_1} \frac{d}{dy} x_{n-1}

    where y_n=\frac{d^n y}{dx^n} and x_n=\frac{d^n x}{dy^n}

    and therefore if I know what each a_n is, I can recursively calculate the coefficients  b_n

    Here's the first few:

    y_1=\frac{1}{x_1}

    y_2=-\frac{x_2}{x_1^3}

    y_3=\frac{3x^2-x_1 x^3}{x_1^5}

    y_4=-\frac{15x_2^3-10 x_1 x_2 x_3+x_1^2 x_4}{x_1^7}

    y_5=\frac{105 x_2^4-105 x_1 x_2^2 x_3+15x_1^2x_2 x^4+10 x_1^2x_3^2-x_1^3 x_5}{x_1^9}

    You guys think it's possible to figure out a non-recursive formula for y_n?

    I can begin to see a trend for n\geq 2:

    y_n=(-1)^{n+1}\frac{\prod_{k=0}^{n-2} (2n+1) x_2^{n-1}+\text{something}+(-1)^{n+1}x_1^{n-2} x_n}{x_1^{2n-1}}
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  2. #2
    Senior Member bkarpuz's Avatar
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    Hi shawsend,

    it seems that you obtain a difference equation (or a recursion), but just as in the theory of differential equations, you can not always solve a difference equation if its not a special type, i.e., linear constant coefficients or ect, either. Therefore, it is not very surprising that you may not have an explicit formula for it. I just wanted to note this.
    But I hope I can find what you want...

    bkarpuz
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  3. #3
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    Quote Originally Posted by bkarpuz View Post
    Hi shawsend,

    it seems that you obtain a difference equation (or a recursion), but just as in the theory of differential equations, you can not always solve a difference equation if its not a special type, i.e., linear constant coefficients or ect, either. Therefore, it is not very surprising that you may not have an explicit formula for it. I just wanted to note this.
    But I hope I can find what you want...

    bkarpuz
    . . . dang it! I hate when that happens. I'm thinking now maybe that's the reason I'm not finding an explicit formula but then, that is no way to do mathematics either; just because no one figured out one is no guarantee there is none.
    Last edited by shawsend; September 13th 2009 at 07:49 AM.
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