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VERY hard integral (atleast to me)

  1. Jul 6, 2005 #1
    I am trying to integrate from 0 to 8

    C (t)e^-rt


    and C(t) is equal to 50t^1/2 (50 times the square root of t)


    I was thinking I could use an infinite series method, the one in the form of e to the x, and my x would be -rt. But if I decide to go that route, then what would I do with 50t^1/2 ?

    What do you all think?
     
  2. jcsd
  3. Jul 6, 2005 #2

    dextercioby

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    Make the sub [itex] u=\sqrt{t} [/itex] and then part integration and use the definition of "erf".

    Daniel.
     
  4. Jul 6, 2005 #3
    Yes I think part integration is the way to go.

    In case you have no idea what erf(x) is, here is a link to read up on.

    [tex]erf(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^2}dt[/tex]

    Jameson
     
    Last edited: Jul 6, 2005
  5. Jul 6, 2005 #4

    dextercioby

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    Your definition is not normalized to unity. We like those kind of definitions, though.

    Daniel.
     
  6. Jul 6, 2005 #5
    I'm sorry, I don't quite understand how a definition is normalized to unity. Please explain. Is the variables, lack of defining things...?
     
  7. Jul 6, 2005 #6

    dextercioby

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    I meant the function. For your "erf" [itex] \lim_{x\rightarrow +\infty} \ erf(x) =\frac{1}{2} [/itex].

    Daniel.

    P.S.You should have picked the constant as to make the limit "+1".
     
  8. Jul 6, 2005 #7
    Apologies then to all. I was just putting my two cents of what I knew in. As usual Daniel, you know much more than myself.
     
  9. Jul 6, 2005 #8

    dextercioby

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    It's not compulsory to use a constant before the integral, u can drop it. But we usually have conventions when defining special functions.

    We can define

    [tex] \tan x=:\sqrt{\pi\sqrt{e\sqrt{\varphi}}} \ \frac{\sin x}{\cos x} [/tex]

    ,but the convention is to choose the constant =1.

    Daniel.
     
  10. Jul 6, 2005 #9
    Could you explain why? Why doesn't the constant make a difference?
     
  11. Jul 6, 2005 #10

    dextercioby

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    Why would it? You can define the function

    [tex] eerf (x) =\int_{0}^{x} \exp\left(-t^{2}\right) \ dt [/tex].

    Daniel.
     
  12. Jul 6, 2005 #11

    dextercioby

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    First of all, you need to address it with a diferent name."Tangent" and the shortening "tan" are already used for a function. Then you need to make your function known and accepted. Since it's basically a rescaling of an already existing object, i'm sure everyone will reject it. As i said, it's all a matter of conventions and definitions. It's the majority of mathematicians that decide whether your convention/definition or "X"'s is better and should be accepted.

    I think this is running off topic and it shouldn't. If i didn't make my point clear, that's it.

    Daniel.
     
  13. Jul 6, 2005 #12
    I see your point. I was just confused if you were discussing new functions or ones previously established.

    From Mathworld - "Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define erf(z) without the leading factor of [itex]\frac{2}{\sqrt{\pi}}[/itex]"

    So it seems both definitions are acceptable.
     
  14. Jul 8, 2005 #13
    For example, if I had my way, sin(x) would be denoted by $(x), and cos(x) would be denoted by ©(x). :biggrin:
     
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