Help with Complex Limit: s, N → ∞

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In summary, the given limit is equal to 2 times the limit as N approaches infinity of the sum of an integral and a constant term. The integral can be simplified by substituting u=sx/2π, and the final expression can be obtained by combining differences of logs and moving the constant term outside the sum.
  • #1
mmzaj
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We have the following limit:
[tex]\lim _{N\rightarrow \infty}N\log\left(1+\frac{(s\log N)^{2}}{4\pi^{2}} \right )-\sum_{n=1}^{N}\log\left(1+\frac{(s\log n)^{2}}{4\pi^{2}} \right )-N\left(\frac{2\log N}{(\log N)^{2}+\frac{4\pi^{2}}{s^{2}}} \right ) [/tex]

Where
png.png
is a complex parameter.

any thoughts are appreciated
 
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  • #2
Is this homework?
Is the third large term inside the sum or outside?
Did you use ##N = \sum_{n=1}^N 1## and combine the differences of logs to a log of fractions?
 
  • #3
this is not a homework
the third term is outside the sum
i tried your suggestion, but wasn't helpful

thanks for the remarks though
 
Last edited:
  • #4
the limit is better stated this way, i guess.
[tex]2\lim_{N\rightarrow \infty}\sum_{n=1}^{N} \left[\int_{\log n}^{\log N}\frac{x}{x^{2}+\frac{4\pi^{2}}{s^{2}}}dx-\left(\frac{\log N}{(\log N)^{2}+\frac{4\pi^{2}}{s^{2}}} \right ) \right ][/tex]
 
  • #5
If we split up the expression, the last part is not dependent of n and can be moved outside the sum. In the integral, if you put u=sx/2π, you get x=2πu/s and therefore dx=2π/s*du. This will make the integral easier to solve. Just remember to change the integration limits (x = log(n) transforms into u=s*log(n)/2π).
 
  • #6
The integral is easy to solve (no substitution necessary, the numerator is 1/2 the derivative of the denominator), but then we are back at the expression in post 1.
 

1. What is a complex limit?

A complex limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. In the case of complex limits, the input and output values are complex numbers, which consist of a real and imaginary component.

2. How is a complex limit different from a regular limit?

A regular limit deals with real numbers, while a complex limit deals with complex numbers. In a regular limit, the input approaches a real number from the left and right sides, while in a complex limit, the input approaches a complex number from any direction in the complex plane.

3. What is s in the complex limit s, N → ∞?

The variable s represents the complex input in the limit, while N represents the number of iterations or steps taken towards the limit. In this case, N represents infinity, meaning the input is approaching a complex number infinitely.

4. How do I solve a complex limit?

To solve a complex limit, you can use techniques such as L'Hopital's rule or the squeeze theorem. It is also important to understand the properties of complex numbers and how they behave in a limit. It is recommended to consult a textbook or seek guidance from a mathematics professional for a thorough understanding.

5. What is the significance of taking the limit as N approaches infinity?

Taking the limit as N approaches infinity allows us to understand the long-term behavior of a function. It helps us determine if the function will approach a specific value, oscillate between values, or diverge to infinity. This can provide valuable insights in various areas of mathematics and science, such as calculus, engineering, and physics.

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