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3.7.1: Series Additional Material

  • Page ID
    84392
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    By Carey A. Smith

    Example \(\PageIndex{1}\) Bessel Function

    The image of a star thru a space telescope, which has no aberrations, is called an Airy pattern, shown in this image:

    Airy Pattern.png

    Figure \(\PageIndex{1}\): Airy pattern

    (https://en.Wikipedia.org/wiki/Airy_disk, This work has been released into the public domain by its author, Sakurambo at English Wikipedia.)

    The formula for a cross-section of the Airy pattern intensity is:

    \[ I(\theta) = I_0 \left[ \frac{2J_1(k*a*sin\theta)}{k*a*sin\theta} \right]^2 = I_0 \left[ \frac{2J_1(x)}{x} \right]^2 \]

    where J1(x) is the 1st-order Bessel function of the 1st kind.

    This Bessel function is approximated by this infinite series with α = 1:

    \[J_\alpha (x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!*\Gamma(m+\alpha+1)} * \left(\frac{x}{2}\right)^{(2m+\alpha)} \]

    For m!, use the Matlab function factorial(m)

    Г(p) = the Gamma function. In Matlab this function is gamma(p).

    When is a positive integer, gamma(p) = (p-1)!

    Instructions:

    1. Set x = 0.5
    2. Write a for loop to sum the terms of this Bessel function series, Jα(x), for m = 0 to 10.
    3. Display the last term
    4. Display the resulting sum

    Solution

    term = 1.5697e-27
    J1_x = 0.24227

    Exercise \(\PageIndex{1}\) atan(x) Taylor's series and absolute error

    The Taylor's series for the arctangent function is:

    atan(x) = x - x3/3 + x5/5 - x7/7 + ...

    The general formula for the kth term is computed with these 2 lines of code:

    m = (2*k-1) % This generates 1, 3, 5, 7, ...

    term = (-1)^(k-1)*x^m / m % (-1)^(k-1) = 1, -1, 1, -1, ...

    Because the sign changes from term to term, this is called an alternating series.

    (1 pt) Write a Matlab m-file script. This script will have a "for loop". The details of the for loop are described below.

    (1 pt) Put these lines of code at the beginning of your file:

    % Compute the Taylor's series for atan(x)
    % Clear any variables; close any figures; eliminate white space; clear the console
    clear all; close all; format compact; clc;
    % Open a figure for plotting the patial sums
    figure;
    hold on;
    grid on;

    (1 pt) Initialize these variables:

    x = pi/5 % Set the x-value
    atan_series = 0; % Initialize the series' sum

    (2 pts) Write a for loop. Let k = the for-loop index. k goes from 1 to 8

    (3 pts) Inside the for loop, compute term as specified above for each iteration.

    Then add term to the previous value of atan_series with this code:

    atan_series = atan_series + term

    (1 pt) Inside the for loop, also plot the kth partial sum with this code:

    plot(k, atan_series,'o');

    (1 pt) After the for loop, display the sum and the last term with these lines of code:

    atan_series % Display the series sum

    term % Display the last term that was calculated

    (1 pt) Compute and display atan_Matlab = atan(x) [Matlab's buit-in function]

    (1 pt) Compute and display the absolute error = abs( atan_series - atan_Matlab)

    If the code is done correctly, the absolute error should be < 0.001

    Answer

    Add texts here. Do not delete this text first.


    This page titled 3.7.1: Series Additional Material is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.

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