[net.math] Power Series Problem

andrews@yale.ARPA (Thomas O. Andrews) (02/20/85)

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Recently, the following problem was posted in net.math:

[Paraphrased - I can't find the original article.]

   Find sequence a1,a2,a3,... such that

	 oo
	----
	\       a     n
  f(x)=  \       n   x
	 /    -----                satisfies f(n)=a .
	/       n!                                 n
	----
	n=0


I believe I've found a nice, 'simple' function f that does the trick.

Let c be a complex root of    c=exp(c).  (Such roots exist.)


Let           n
	a  = c  .
	 n

Then f(x)= {summation above} = exp(cx) .  Hence f(n)=exp(cn).

			    n
But exp(c)=c, so exp(cn)=  c = a
				n.

We can bet a solution of this problem with the a 's real by taking
						n
	 __            ____
b = a  + a      (Where x+yi =x-yi.)
 n   n    n

Then we get
		 _
f(x)=exp(cx)+exp(cx).   In particular, if c=u+vi, then we can write

f(x)=exp(ux) (exp(vxi)+exp(-vxi)) = 2*exp(ux)*cos(vx).


-- 
					      Thomas Andrews

Have you killed a yellow pig today?