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Math Help - Construct a function satisfied some conditions

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    Thumbs up Construct a function satisfied some conditions

    Construct a function f(x)\in C^{\infty}(-\infty,+\infty)
    that satisfy f^{(2n+1)}(0) = n and f^{(2n)}(0) = 0,for all n\geq0.
    Furthermore,show that there are infinity f(x)satisfy the condition above
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    Quote Originally Posted by Xingyuan View Post
    Construct a function f(x)\in C^{\infty}(-\infty,+\infty)
    that satisfy f^{(2n+1)}(0) = n and f^{(2n)}(0) = 0,for all n\geq0.
    Furthermore,show that there are infinity f(x)satisfy the condition above
    The function \sum_{n=0}^\infty\frac{nx^{2n+1}}{(2n+1)!} = \tfrac12(x\cosh x-\sinh x) will have that property. So will any function obtained from that one by adding any multiple of a C^\infty-function whose derivatives at 0 are all 0, such as the function f(x) = \begin{cases}e^{-1/x^2}&(x\ne0),\\0&(x=0).\end{cases}
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