VERY hard integral (atleast to me)

[Techman07]

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?

 

[dextercioby]

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

Daniel.

 

[Jameson]

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.

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

Jameson

 

[dextercioby]

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

Daniel.

 

[Jameson]

I'm sorry, I don't quite understand how a definition is normalized to unity. Please explain. Is the variables, lack of defining things...?

 

[dextercioby]

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

Daniel.

P.S.You should have picked the constant as to make the limit "+1".

 

[Jameson]

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.

 

[dextercioby]

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

$$\tan x=:\sqrt{\pi\sqrt{e\sqrt{\varphi}}} \ \frac{\sin x}{\cos x}$$

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

Daniel.

 

[Jameson]

Could you explain why? Why doesn't the constant make a difference?

 

[dextercioby]

Why would it? You can define the function

$$eerf (x) =\int_{0}^{x} \exp\left(-t^{2}\right) \ dt$$.

Daniel.

 

[dextercioby]

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.

 

[Jameson]

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 $\frac{2}{\sqrt{\pi}}$"

So it seems both definitions are acceptable.

 

[Manchot]

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).