Revisions to Approximation of Spherical Bessel function [closed]

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edited Jun 19, 2022 at 11:32

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I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a Mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the Mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}}. \end{equation} I also have to use the derivative of the Bessel function which they approximate as \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2}. \end{equation}

IThis approximations are done in the limit where both $x$ and $xl$ are large. I would very much appreciate anyone bringing some insights the derive these approximations!

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a Mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the Mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}}. \end{equation} I also have to use the derivative of the Bessel function which they approximate as \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2}. \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a Mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the Mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}}. \end{equation} I also have to use the derivative of the Bessel function which they approximate as \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2}. \end{equation}

This approximations are done in the limit where both $x$ and $xl$ are large. I would very much appreciate anyone bringing some insights the derive these approximations!

Post Closed as "Needs details or clarity" by ZeroTheHero, Jon Custer, Michael Seifert, Níckolas Alves, GiorgioP-DoomsdayClockIsAt-90

occurred Jun 18, 2022 at 5:18

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edited Jun 17, 2022 at 0:34

Níckolas Alves

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I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a mathematicaMathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the mathematicaMathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is: \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}} \end{equation}\begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}}. \end{equation} I also have to use the derivative of the Bessel function which they approximate in the following way:as \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2} \end{equation}\begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2}. \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is: \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}} \end{equation} I also have to use the derivative of the Bessel function which they approximate in the following way: \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2} \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a Mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the Mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}}. \end{equation} I also have to use the derivative of the Bessel function which they approximate as \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2}. \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!

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edited Jun 10, 2022 at 9:43

god_operator

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I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution, of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is: \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}} \end{equation} I also have to use the derivative of the Bessel function which they approximate in the following way: \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2} \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution, of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is: \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}} \end{equation} I also have to use the derivative of the Bessel function which they approximate in the following way: \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2} \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a mathematica notebook that I am following, they work with spherical Bessel functions in order to free stream the multipole solution of the fluid equations in Fourier space. I understand the analytical implementation of the Bessel functions in the formula, but in the mathematica code, they approximate these functions in a way which I have not been able to derive for myself or find online. The approximation of the Bessel function is: \begin{equation} l^2 j_l^2(xl) = \frac{1}{2x\sqrt{x^2-1}} \end{equation} I also have to use the derivative of the Bessel function which they approximate in the following way: \begin{equation} l^2 j_l'^2(xl) = \frac{1}{2x\sqrt{x^2-1}}\frac{x^2-1}{x^2} \end{equation}

I would very much appreciate anyone bringing some insights the derive these approximations!