I am trying to find the upper bound of $$ \sum_{n \geq 1} J_0(an) J_0(bn) \sin(cn) \sin(dn) $$ where $J_0(x)$ is the zeroth order Bessel function of first kind, and $a,b \geq 0, \textit{ and } c,d \in \mathbb{R}$. I have tried to use Poisson summation formula but without any success. Any kind of help is welcome. Thank you.
2026-02-23 08:59:24.1771837164
Infinite series Sum of zeroth order Bessel Functions of first kind
159 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in FOURIER-SERIES
- order of zero of modular form from it's expansion at infinity
- Fourier series expansion of $\frac{\pi^4}{96}$ and $\frac{\pi^4}{90}$
- How is $\int_{-T_0/2}^{+T_0/2} \delta(t) \cos(n\omega_0 t)dt=1$ and $\int_{-T_0/2}^{+T_0/2} \delta(t) \sin(n\omega_0 t)=0$?
- Fourier series. Find the sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n+1}$
- How get a good approximation of integrals involving the gamma function, exponentials and the fractional part?
- The convolution theorem for fourier series.:$ \widehat{f*g}(x) =2π\hat{g}(x)\cdot\hat{f}(x) $
- Ergodicity of a skew product
- Fourier Series on $L^1\left(\left[0,1\right)\right)\cap C\left(\left[0,1\right)\right)$
- Parseval's Identity Proof Monotone/Dominated Convergence Theorem
- How can I interchange the sum signs
Related Questions in FOURIER-TRANSFORM
- Proof of Fourier transform of cos$2\pi ft$
- Find the convergence of series of a sequence of functions in $L^2(\mathbb{R})$
- solving a simple ODE with Fourier transform
- How can we prove that $e^{-jωn}$ converges at $0$ while n -> infinity?
- Show that a periodic function $f(t)$ with period $T$ can be written as $ f(t) = f_T (t) \star \frac{1}{T} \text{comb}\bigg(\frac{t}{T}\bigg) $
- Taking the Discrete Inverse Fourier Transform of a Continuous Forward Transform
- Arcsin of a number greater than one
- Complex numbers in programming
- Power spectrum of field over an arbitrarily-shaped country
- Computing an inverse Fourier Transform / Solving the free particle Schrödinger equation with a gaussian wave packet as initial condition
Related Questions in BESSEL-FUNCTIONS
- How to prove $\int_{0}^{\infty} \sqrt{x} J_{0}(x)dx = \sqrt{2} \frac{\Gamma(3/4)}{\Gamma(1/4)}$
- What can be said about the series $\sum_{n=1}^{\infty} \left[ \frac{1}{n} - \frac{1}{\sqrt{ n^2 + x^2 }} \right]$
- A closed-form of an integral containing Bessel's function
- Sources for specific identities of spherical Bessel functions and spherical harmonics
- The solution to the integral $\int_{0}^{\infty} \log(x) K_{0}(2\sqrt{x})\,dx$
- Laplace transform of $t^\mu I_\nu(at)$
- Integral of product of Bessel functions of first kind and different order and argument
- Series involving zeros of Bessel functions
- Finding the kernel of a linear map gotten from a linear map with one kind of bessel function $j_i$ and replacing them with the $y_j$
- Transcendental equation with Bessel function
Related Questions in HYPERGEOMETRIC-FUNCTION
- Probability of future success when trials are dependent: In 4 draws w/o replacement, what is the probability that draws 3 & 4 will be successful?
- Integral of a Gaussian multiplied with a Confluent Hypergeometric Function?
- Integral $ \int_{0}^{D} x \cdot (\sqrt{1+x^a})^{b}\,dx$
- Generating function of the sequence $\binom{2n}{n}^3H_n$
- Integral $\int (t^2 - 1)^{a} \cdot t^{b} \cdot \log(t)\,dt$
- The integral of an elliptic integral: $\int_{0}^{1}\frac{x\mathbf{K}^2\left ( x \right )}{\sqrt{1-x^{2}}}\mathrm{d}x$
- Integrating $\int (t^2+7)^{\frac{1}{3}}$dt
- Evaluating a definite integral with hypergeometric function
- Consider a sequence of n independent multinomial trials
- Asymptotic expansion of the hypergeometric function
Related Questions in POISSON-SUMMATION-FORMULA
- Understanding Proof of Poisson Summation Formula
- Evaluating $\displaystyle\int_{-\infty+iy}^{\infty+iy}(cv)^{-k}e\left(\frac{-m}{c^2v}-nv\right)dv$.
- Probability density of the fractional part of |σZ| as σ→∞?
- Minkowski's Theorem by Harmonic Analysis
- G(k,X) is a modular form of weight k and character X
- Proving the Partial Fraction Decomposition of the Hyperbolic Cotangent Function by using Poisson Summation
- Proving the relation: $\sum_{n\in\Bbb Z} \frac{2a}{a^2+4 \pi^2 n^2} = \sum_{n\in\Bbb Z} e^{-a \left\lvert n \right\rvert}$
- Evaluation of a sum by means of Poisson sum formula and digamma function
- Confusion about applying Poisson's Sum Formula
- Question on Poisson summation formula and Fourier transform
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?