Incomplete Elliptic Integrals of the second kind -- evaluation

Discussion in 'Differentiation and Integration' started by Sequoia, Apr 6, 2020.

  1. Sequoia


    Apr 6, 2020
    Likes Received:
    I am programming as a hobby and trying something with elliptical orbits of satellites.
    I need to understand how to measure the length of an arc on an ellipse, e.g. from 0 to pi/3.
    Now I've come upon Elliptic Integrals and an associated brick wall.

    For the "Complete Elliptic Integral of the second kind" the length of an arc from 0 to pi/2, I've found a series the program can use to approximate the value of this kind of arc.

    However, for the INCOMPLETE sort, there is nowhere a description of how the value is calculated, no series, no formula. Nothing.

    They tell you in most places either :
    1. Look it up
    2. use a website to calculate it, such as

    The website can indeed calculate the value where k = 0.6 and the limits are 0 to pi/3 bringing the value:
    0.98922159...... so there must be an infinite series one utilizes to calculate this.

    It is most certainly not an elementary function. So, what is the series?
    Sequoia, Apr 6, 2020
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