A CONJECTURE (could someone help ? )

  • #1
391
0
i have the following conjecture about infinite power series

let be a function f(x) analytic so it can be expanded into a power series

[tex] f(x)= \sum_{n=0}^{\infty} a_{2n}(-1)^{n} x^{2n} [/tex]

with [tex] a_{2n} = \int_{-c}^{c}dx w(x)x^{2n} [/tex] [tex] w(x) \ge 0 [/tex] and [tex] w(x)=w(-x) [/tex] on the whole interval (-c,c) in case c is inifinite this W(x) is ALWAYS positive for every real 'x'

here 'c' is a Real constant , also we can have [tex] c=\infty [/tex]

and [tex] a(n)=0 [/tex] for n=1,3,5,7,9,.. 2n-1

the [tex] a(2n) [/tex] are ALL positive and NONZERO

then f(x) has ONLY real roots or f(x) has NO roots at all (not real or complex)

amazingly it seems true , for example it holds for [tex] sin(x)/x [/tex] the Bessel function of zeroth order [tex] J_0 (x) [/tex] and so on
 
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Answers and Replies

  • #2
336
0
i have the following conjecture about infinite power series

let be a function f(x) analytic so it can be expanded into a power series

[tex] f(x)= \sum_{n=0}^{\infty} a_{2n}(-1)^{n} x^{2n} [/tex]

with [tex] w(x) \ge 0 [/tex] and [tex] w(x)=w(-x) [/tex] on the whole interval (-c,c) in case c is inifinite this W(x) is ALWAYS positive for every real 'x'

here 'c' is a Real constant , also we can have [tex] c=\infty [/tex]

and [tex] a(n)=0 [/tex] for n=1,3,5,7,9,.. 2n-1

the [tex] a(2n) [/tex] are ALL positive and NONZERO

then f(x) has ONLY real roots or f(x) has NO roots at all (not real or complex)

amazingly it seems true , for example it holds for [tex] sin(x)/x [/tex] the Bessel function of zeroth order [tex] J_0 (x) [/tex] and so on
The series simplifies to 2 *integral( 0 to c) {1 / 1+ (tx)^2}w(t) dt , which is posotive for all
real x. I'm not sure about the complex case, though.
 

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