Integrating Over Real Numbers: Understanding the Notation and Limitations

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In summary, the conversation discusses a potential change of variable for the integral \int_0^{\infty}F(x)dx with the limit x\rightarrow\infty F(x)\rightarrow0. The proposed substitution of x=-e^{t} leads to a new integral \int_{-\infty}^{\infty+i\pi}F(-e^t)e^{t}dt, but there is confusion about the bounds of the integral and whether the substitution is valid. The conversation also mentions another potential change of variable involving complex numbers, but it is also uncertain if this substitution is valid. The speaker recommends consulting a book on complex integration for guidance.
  • #1
eljose
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let be the integral:

[tex]\int_0^{\infty}F(x)dx [/tex] with [tex]x\rightarrow\infty F(x)\rightarrow0 [/tex] then we make the change of variable x=-e^{t} then the new integral would become [tex]\int_{-\infty}^{\infty+i\pi}F(-e^t)e^{t}dt[/tex] my question is if we can ignore the integral from [tex] (\infty,\infty+i\pi) [/tex] so we have only the integral [tex]\int_{-\infty}^{\infty}F(-e^t)e^{t}dt [/tex] as for big value the F(x) tends to 0
 
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  • #2
You cannot make this substitution. Note that x > 0, and et is also always greater than 0, so how could x = -et ever be true?
 
  • #3
why not? the same would happen with x=-1/(t+1) x>0 but t<0
 
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  • #4
How can a negative quantity equal a positive quantity?
 
  • #5
i think that you lost a negative sign when making your substitution.

the example you give in post #3 is correct for x>0 t<-1

so it sounds like the unresolved question is your bounds on the integral. I'm not sure how to handle this, but keep two things in mind:

1) remember that you are dealing with a formal limit

[tex]\lim_{a \rightarrow -\infty} \int_{a}^{i\pi} f(x) dx + \lim_{b \rightarrow \infty} \int_{i\pi}^{b} f(x) dx[/tex]

2) you are integrating over [tex]\mathbb{C}[/tex] so you need to get a book on complex integration and see what to do. i suspect that solving the above and integrating as usual will suffice.
 
  • #6
Another question let,s suppose we have the integral:

[tex]\int_{-\infty}^{\infty}F(x)dx [/tex] and make the change of variable x=t+ai with i
=sqrt(-1) then what would be the new limits?..thanks...
 
  • #7
I haven't taken any courses on complex analysis, so maybe I'm unfamiliar with the notation. Over what region are you integrating when you take:

[tex]\int _{-\infty} ^{\infty} F(x)dx[/tex]

To me, that suggests that you're integrating over all the real numbers, i.e. x ranges over the reals. If this is the case, then you again run into the problem of equating x with t + ai because unless t = b - ai for some real b, then x will be never have a real part, and t+ai sometimes will, so the two cannot be equated.
 

1. What is the purpose of the "X" in the equation X=^-e^{t} integral?

The "X" in this equation represents the dependent variable, which is the quantity being measured or calculated in relation to the independent variable, t. In this case, the integral of the exponential function e^t is being evaluated.

2. How is the integral of e^t evaluated in this equation?

The integral of e^t is evaluated using the power rule of integration, which states that the integral of x^n is equal to x^(n+1)/(n+1). In this case, we have e^t, so the integral is equal to e^t/(1+1) = e^t/2.

3. What is the significance of the negative exponent in this equation?

The negative exponent in this equation indicates that the integral is being evaluated from negative infinity to t, rather than from 0 to t. This means that the value of the integral is dependent on the value of t and approaches 0 as t approaches infinity.

4. Can this equation be applied to other functions besides e^t?

Yes, this equation can be applied to any function that can be integrated using the power rule. For example, if the function was x^2, the integral would be x^3/3. The same principle applies to any function with a variable exponent.

5. How is this equation used in scientific research or applications?

This equation is commonly used in physics, engineering, and other scientific fields to model exponential growth or decay processes. It can also be used to solve differential equations and to calculate areas under exponential curves.

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