1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Euler' s formula in correct ?

  1. Nov 28, 2008 #1
    1. I use Euler formula of arctan to calculate arcsine



    2. This equation[tex]\arctan x = \frac{x}{1+x^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k x^2}{(2k+1)(1+x^2)}.[/tex]



    If I input 0.9999999999, I will not be able to get the expected result. If input = 0.9 then it is pretty correct, but 0.99 is definitely wrong and more wrong when the decimal digits get higher till uncomputable.
    Could someone help me ? :wink:
     
  2. jcsd
  3. Dec 8, 2008 #2
    2 weeks already and everyone stopped breathing into my thread!
    you ignore my post I guess!
     
  4. Dec 8, 2008 #3

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    It sounds like you just need a better calculator, try using Mathematica.
     
  5. Dec 8, 2008 #4

    Avodyne

    User Avatar
    Science Advisor

    It wasn't clear what you were asking. It's an infinite series, so you must be truncating it at some point, and so of course the result is not exact.
     
  6. Dec 8, 2008 #5
    I don't see what the problem is, I get at least 4 digits correct If I evaluate the sum for n<=13 and 10 digits correct for n<=32. (I don't think n=0 should count).
    Because [tex] \frac{2k x^2}{(2k+1)(1+x^2)} [/tex] is smaller than 1/2 for all x, the error must halve for each increase of n.

    I used the following python program

    Code (Text):
    rom math import *

    x = 0.9999
    limit = 20

    sum = 0
    for n in range (1, limit):
        prod = 1
        for k in range (1, n):
            prod *= 2*k*x*x/((2*k+1)*(1+x*x))
        sum += prod
        print n, atan(x), sum * x / (1+x*x)

    1 0.785348160897 0.4999999975
    2 0.785348160897 0.666649995833
    3 0.785348160897 0.733303328833
    4 0.785348160897 0.761866186262
    5 0.785348160897 0.77455952004
    6 0.785348160897 0.780328640213
    7 0.785348160897 0.782991044782
    8 0.785348160897 0.784233375995
    9 0.785348160897 0.784817943983
    10 0.785348160897 0.785094816918
    11 0.785348160897 0.785226647987
    12 0.785348160897 0.785289691324
    13 0.785348160897 0.785319949099
    14 0.785348160897 0.7853345162
    15 0.785348160897 0.785341547891
    16 0.785348160897 0.785344949982
    17 0.785348160897 0.785346599315
    18 0.785348160897 0.78534740034
    19 0.785348160897 0.785347789989
    20 0.785348160897 0.785347979799

    Converges nicely as you can see.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Euler' s formula in correct ?
  1. Euler's formula eiθ (Replies: 5)

  2. Eulers Formula variant (Replies: 1)

  3. Using Euler's formula (Replies: 3)

Loading...