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I think I broke Wolfram?

  1. Aug 24, 2010 #1
    I think I broke Wolfram?!?!

    1. The problem statement, all variables and given/known data

    I was trying rewrite properly the tangent and arctangent function and was trying to figure out what values of the argument in the natural log was < 0 so that way I could simplify my formulas further because

    i ln(-1) = -pi/2
    and stuff of the sort

    (-x i)/SQRT(x^2 + 1) + SQRT(x^2 +1)/(x^2 + 1) < 0
    basically looking for the first x value that would make that a negative number

    if you have no idea what I'm talking about don't worry about just tell me why I don't get a solution

    http://www.wolframalpha.com/input/?i=solve(-x+i)/SQRT(x^2+++1)+++SQRT(x^2++1)/(x^2+++1)+<+0

    go to that site

    as you can see it didn't give me a solution
    wolfram is suppose to say "no solution exists" when there's not one so what does this mean when it doesn't tell me anything? Did I break it? Is there a solution?

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Aug 24, 2010
  2. jcsd
  3. Aug 24, 2010 #2

    CompuChip

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    Re: I think I broke Wolfram?!?!

    I ran your expression through Mathematica, here is a plot of its real and imaginary parts.
    It looks like Wolfram Alpha is right, and the real part only tends to zero asymptotically.
     

    Attached Files:

  4. Aug 24, 2010 #3
    Re: I think I broke Wolfram?!?!

    ok is there an answer for what is the first negative number that makes the equation negative becasue i need to know that

    (-x i)/SQRT(x^2 + 1) + SQRT(x^2 +1)/(x^2 + 1) < 0

    how do i do this then

    like i'm looking for the x value that will make that expression negative... what's the first value that will do this? Is there a way to figuer this out
     
  5. Aug 24, 2010 #4

    CompuChip

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    Re: I think I broke Wolfram?!?!

    That is my point... I don't know if you can see my attachment yet, but it clearly shows that
    a) the imaginary part is non-zero whenever x is non-zero, so the comparison "Expression < 0" is useless anyway, you can at most speak about "Real part of expression < 0"
    b) the real part is always positive, and only tends to zero as |x| goes to infinity (if you can't see my picture, you can ask Wolfram Alpha)

    [edit]Actually that last link shows you that the real part of your expression is 1/sqrt(x2 + 1) which is non-zero for all real and even complex x.
     
  6. Aug 24, 2010 #5
    Re: I think I broke Wolfram?!?!

    hmmm so that expression can never be negative? hmm i think you uploaded the wrong picture lol as in the picture...

    anways ya ok it can never be negative but what about
    +/- because sqare roots have two solutions
    so i would get a negative hmmmm
     
  7. Aug 24, 2010 #6
  8. Aug 24, 2010 #7
    Re: I think I broke Wolfram?!?!

    When you put in an expression that doesn't make sense you get useless results. Your first problem was trying to order a complex number. What exactly does < mean in the complex plane?
     
  9. Aug 24, 2010 #8
    Re: I think I broke Wolfram?!?!

    Guess you have a point there ok but I still need to figure out the smallest x value that would make that expression negative so that way I can simplify further and state the domain of my new equation that it applies to
     
  10. Aug 24, 2010 #9
    Re: I think I broke Wolfram?!?!

    Your original expression is always a complex number for real x. So it is never negative.
    The expression here : http://m.wolframalpha.com/input/?i=solve%28-x+i%29%2F-SQRT%28x%5E2+%2B+1%29+-SQRT%28x%5E2+%2B1%29%2F%28x%5E2+%2B+1%29+%3D-1 is different from what you first posted; there is a negative sign on the second term.
     
  11. Aug 24, 2010 #10
    Re: I think I broke Wolfram?!?!

    Ya because square roots have two solutions one positive and one negative? I just left the positive and negative symbol +/- out or was I just not suppose to put in to begin with I thought I was suppose to but left it out in the first expression.... ? If you want I can provide some background on where that expression came from let me know
     
  12. Aug 24, 2010 #11
    Re: I think I broke Wolfram?!?!

    Square roots have positive and negative sign when you are trying to solve equations such as [tex]a^{2}= b[/tex]. When a equation involving square roots is defined it is not up to you to include a plus or minus. You only include them if they are a result of an equation involving even roots. Anyway the expressions in the square roots in your question are always positive. In your first expression the term to the right is always positive and greater than zero. The expression to the left is a complex number whose imaginary part is < = 0.
     
  13. Aug 24, 2010 #12
    Re: I think I broke Wolfram?!?!

    hmmm ok well thanks...
    So I think they do have positive and negative signs because I was dealing with an equation like that let me explain

    I started out deriving a better equation for tan(x)

    got this

    tan(x) = (1- e^(-i 2 x) )/(i + i e ^(-i 2 x) ) + pi n

    I checked in my calculator and I think it's right

    I then said well for arctangent
    what does that equal
    tan^-1 (x) = i ln ( (-xi)/SQRT(x^2 + 1) + SQRT(x^2 + 1)/(x^2 + 1) ) + pi n
    checked in my calculator I think it's right

    third I was like what if I knew tan(x) = -3 or some other number how would I solve for the angle... well then it would have two solutions because you wouldn't obey the restrictions when you sued this form of arctangent(-3) because it's an equation not just simply arctan(-3)

    I got the following answers

    i ln ( (-xi)/SQRT(x^2 + 1) + SQRT(x^2 + 1)/(x^2 + 1) ) + pi n
    and
    i ln ( (xi)/SQRT(x^2 + 1) - SQRT(x^2 + 1)/(x^2 + 1) ) + pi n

    checked in my calculator believe it's right

    so as you can see I was wondering if I could simplify this further because of what I know of complex nubmers and the complex logarithm and i ln(i) can be simplified and iln(-1) can as well so that was why I asked that question

    so in this case then wouldn't I need to have the +/- sign in front of the SQRTS I can't remember...
     
    Last edited: Aug 24, 2010
  14. Aug 24, 2010 #13
    Re: I think I broke Wolfram?!?!

    You realize the equation you wrote down is not for tan(x) and tan^-1(x) ?

    They are for tan(z) and tan^-1(z).

    EDIT

    Where did you even get those equations from ? I'm not sure if they are correct.

    When x=0 you get tan(0) = pi n ?
     
    Last edited: Aug 24, 2010
  15. Aug 24, 2010 #14
    Re: I think I broke Wolfram?!?!

    ya I do realize that... ok so then just take the pi n
    off of
    tan(x) = (1 - e^(-2ix) )/(i + i e^(-2 i x) )
    there?
     
  16. Aug 24, 2010 #15
    Re: I think I broke Wolfram?!?!

    And I actually thought they applied to all reals as well because it works fine as well when you use noraml angle measures right like pi/3 pi/2 etc.?

    B.T.W. I started out with sine and cosine and used the fact that tangent equals sine/cosine and simplified that is all

    B.T.W. thanks for pointing that out for me I forgot that you don't put + pi n for the normal formulas of tangent, sine, cosine, etc. but for tan^-1 and stuff you do... Sorry I'm tired...
     
  17. Aug 24, 2010 #16
    Re: I think I broke Wolfram?!?!

    EDIT

    This equation is correct, sorry about that.
     
    Last edited: Aug 24, 2010
  18. Aug 24, 2010 #17
    Re: I think I broke Wolfram?!?!

    sin(x) = (e^(ix) - e^(-ix))/(2i)
    cos(x) = (e^(ix) + e^(-ix))/2
    tan(x) = sin(x)/cos(x)
    =( (e^(ix) - e^(-ix))/(2i) ) /( (e^(ix) + e^(-ix))/2 )
    = ( 2( (e^(ix) - e^(-ix) ) )/ (2i (e^(ix) + e^(-ix)) )
    = (e^(ix) - e^(-ix) )/(i((e^(ix) + e^(-ix)))
    = (e^(ix)(1 - e^(-2ix)))/(ie^(-ix)(1 + e^(-2ix) ) )
    = (1 - e^(-2ix) )/(i (1 + e^(-2ix) ) )
    = (1 - e^(-2ix) )/(i + i e^(-2ix) )
    ?

    this will be easier to read
    cis(x) = e^(ix) = cos(x) + i sin(x)
    cis(-x) = e^(-ix) = cos(x) - i sin(x) = 1/e^(ix)

    sin(x) = (cis(x) - cis(-x) )/(2i)
    cos(x) = (cis(x) + cis(-x) )/2
    tan(x) = sin(x)/cos(x)
    =( (cis(x) - cis(-x) )/(2i) )/( (cis(x) + cis(-x) )/2 )
    = ( 2(cis(x) - cis(-x)) )/( 2i(cis(x) + cis(-x)) )
    = ( (cis(x) - cis(-x) )/( i(cis(x) + cis(-x)) )
    =( cis(x)(1 - cis^2(-x)) )/( i cis(x)(1 + cis^2(-x)) )
    =( 1 - cis^2(-x) )/( i(1 + cis^2(-x) )
    =( 1 - cis^2(-x) )/( i + i cis^2(-x) )
    =( 1 - e^(-i2x) )/(i + i e^(-i2x) )

    and if you really wanted
    =( 1 - 1/e^(i2x) )/(i + i/e^(2ix)
    or expressed this way if you wanted I guess
    =(1 - 1/cis^2(x) )/(i + i/cis^2(x)

    ?

    done

    I don't see what I've done wrong or were I went wrong if i did =(
     
    Last edited: Aug 24, 2010
  19. Aug 24, 2010 #18
    Re: I think I broke Wolfram?!?!

    Sorry, you are correct.

    Okay, can you show your other steps?

    Like how you derived arctan(z) ?
     
  20. Aug 24, 2010 #19
    Re: I think I broke Wolfram?!?!

    [tex]arctan(z)=\frac{1}{2}log\frac{i+z}{i-z}[/tex]

    Show me how you got your own formula.
     
  21. Aug 24, 2010 #20
    Re: I think I broke Wolfram?!?!

    Then I was like ok if

    tan(x) = ( 1 - e^(-i2x) )/(i + i e^(-i2x) )
    then what if I know that tan(x) = #
    like for example tan(pi/4) = SQRT(2)/2
    I was like well what if I knew this instead and didn't know the angle
    tan(x) = SQRT(2)/2
    then I would have to do this to solve
    tan(SQRT(2)/2)
    and what if I didn't know what this was equal to? what if I didn't know it was pi/4 hmmm I could just use a calculator but that is cheating =)
    so I said ok then

    tan(x) = ( 1 - e^(-i2x) )/(i + i e^(-i2x) ) = #
    and solved for x the angle and go the solutions
    i ln ( (-xi)/SQRT(x^2 + 1) + SQRT(x^2 + 1)/(x^2 + 1) ) + pi n
    and
    i ln ( (xi)/SQRT(x^2 + 1) - SQRT(x^2 + 1)/(x^2 + 1) ) + pi n

    I think there right
    I also figured out that if it was just tan^-1(x) and had to obey the restrictions it would just be this equation only

    i ln ( (-xi)/SQRT(x^2 + 1) + SQRT(x^2 + 1)/(x^2 + 1) ) + pi n
     
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