VERY hard integral (atleast to me)

In summary, the conversation discusses the integration of a function from 0 to 8, with C(t) being equal to 50 times the square root of t. The use of an infinite series method is suggested, as well as making the substitution u=\sqrt{t} and using part integration and the definition of erf(x). The discussion also touches on the normalization of the function and the importance of conventions and definitions in mathematics.
  • #1
Techman07
12
0
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?
 
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  • #2
Make the sub [itex] u=\sqrt{t} [/itex] and then part integration and use the definition of "erf".

Daniel.
 
  • #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
 
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  • #4
Your definition is not normalized to unity. We like those kind of definitions, though.

Daniel.
 
  • #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...?
 
  • #6
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".
 
  • #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.
 
  • #8
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.
 
  • #9
Could you explain why? Why doesn't the constant make a difference?
 
  • #10
Why would it? You can define the function

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

Daniel.
 
  • #11
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.
 
  • #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.
 
  • #13
dextercioby said:
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.
For example, if I had my way, sin(x) would be denoted by $(x), and cos(x) would be denoted by ©(x). :biggrin:
 

What is a "VERY hard integral"?

A "VERY hard integral" is a type of mathematical problem that involves finding the area under a curve of a complex function. It is considered "very hard" because it requires advanced mathematical knowledge and techniques to solve.

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