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Results tagged with real-analysis
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user 145223
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
5
votes
0
answers
650
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here …
- 774
9
votes
0
answers
506
views
On Riesz criteria for Riemann hypothesis:
Marcel Riesz defined a function :
$R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$
The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$)
For any $\varepsilon$
We have …
- 774
15
votes
1
answer
656
views
Fourier's proof of reality of all roots of Bessel function $J_0(x)$
In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real.
I want to ask if there is a modern version of this proof exist in literature?
If someone can provide …
- 774