A CONJECTURE (could someone help ? )

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In summary, the conjecture is that for a function f(x) that can be expanded into a power series, as long as certain conditions are met, it will have either only real roots or no roots at all. This has been observed in examples such as the Bessel function of zeroth order.
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zetafunction
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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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  • #2
zetafunction said:
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.
 

1. What is a conjecture?

A conjecture is a statement that is believed to be true but has not yet been proven. It is essentially an educated guess or hypothesis that requires further investigation and evidence to be confirmed.

2. How is a conjecture different from a theorem?

A theorem is a statement that has been rigorously proven using mathematical proof, while a conjecture is an unproven statement that is based on observation, experimentation, and/or mathematical intuition.

3. Can a conjecture be proven?

Yes, a conjecture can be proven if there is enough evidence and logical reasoning to support it. However, some conjectures may never be proven and remain as open problems in mathematics.

4. What is the importance of conjectures in science?

Conjectures play a crucial role in the advancement of science as they provide a starting point for further research and investigation. They also help scientists to make predictions and test their theories, leading to new discoveries and advancements in the field.

5. How do scientists test a conjecture?

Scientists test conjectures by using mathematical reasoning, conducting experiments, and analyzing data. They also seek feedback and critiques from other experts in the field to refine and validate their conjectures.

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