Re: CMS: where are the zeroes ?

On 29 apr, 21:18, umum...@xxxxxxxxx wrote:

On 26 apr, 20:44, umum...@xxxxxxxxx wrote:

In real analysis, we have the following truncated series:

P_n(x) = 1 + x + x^2/2 + x^3/3! + .. + x^n/n!

And there exists a remainder term  R_n(x) = x^(n+1)/(n+1)!.exp(t.x)
where t is a real number between 0 and 1, such that

  exp(x) - P_n(x) - R_n(x) = 0

It's easy to see that the function [ exp(z) - P_n(z) - R_n(z) ] is
entire and that it is zero on the real axis. We conclude [ .. ]
that it is zero in the whole complex plane. Therefore:

  exp(z) - P_n(z) = R_n(z)

Now I see that this is not true either. The reason is that t is
not independent of x , it is a function of x . Thus it is not at
all clear how this would extend to the complex plane. Right ?

  | exp(z) - P_n(z) | = | z^(n+1)/(n+1)!.exp(t.z) |
                      = |z|^(n+1)/(n+1)!.|exp(z)|^t
                      < |z|^(n+1)/(n+1)!.|exp(z)|

Here it goes wrong. The above is only true if  | exp(z) | > 1 ,
which is only true for  Re(z) > 0 (while the region for  Re(z) < 0
is the critical one). It makes the rest of my argument invalid.

Han de Bruijn
.

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