# Thread: Construct a function satisfied some conditions

1. ## Construct a function satisfied some conditions

Construct a function $\displaystyle f(x)\in C^{\infty}(-\infty,+\infty)$
that satisfy $\displaystyle f^{(2n+1)}(0) = n$ and $\displaystyle f^{(2n)}(0) = 0$,for all $\displaystyle n\geq0$.
Furthermore,show that there are infinity $\displaystyle f(x)$satisfy the condition above

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