Integral of 1/x in complex variables

In summary, the conversation discusses the difference between integrals in the complex plane and the real plane, specifically regarding the absolute value restraint in ln|x| and whether it can be relaxed to ln(x). The use of principal branch of the logarithm in the complex plane is also mentioned, as well as the existence of a branch where the complex logarithm is not holomorphic.
  • #1
Simfish
Gold Member
823
2
Does [tex]\int 1/x dx = ln|x| + c[/tex] in complex variable theory? Or can we relax the absolute value restraint of ln|x|? (as in, can it be ln(x) + c?)
 
Physics news on Phys.org
  • #2
I'm not sure integrals work exactly the same in the complex plane, I highly doubt it but I haven't quite reached there in my textbook. However, choosing the principal branch of the log in the complex plane, we know from Euler that [tex]e^{i\pi}= -1[/tex] So the logs of negative numbers are quite easy to extend from the reals: [tex]\log (-b) = \log b + \log (-1) = \log b + i\pi[/tex]
 
  • #3
not if you are integrating over a closed path containing the origin. Although this is a correct answer it doesn't give much insight to your question..complex integration is significantly different from it's real analogue.

for example: lnx as a function is senseless unless you've defined lnx as the complex logarithm, which is actually pretty complicated as GibZ's msg above suggests.
 
  • #4
It is true that if [tex]\log z[/tex] is any logarithm along some branch B. Then [tex](\log z)' = 1//z[/tex] for all values not on B. No matter how you choose to define the complex logarithm there will be a branch where it will not be holomorphic. But it does exist at least partially.
 

1. What is the definition of the integral of 1/x in complex variables?

The integral of 1/x in complex variables is a mathematical concept that represents the area under the curve of the function 1/x in the complex plane. It is denoted by ∫(1/x)dz, where dz is a small element in the complex plane.

2. Is the integral of 1/x in complex variables well-defined?

No, the integral of 1/x in complex variables is not well-defined. This is because the function 1/x has a singularity at the point z = 0, which makes the integral diverge. Therefore, we need to consider the integral over a path that avoids this singularity in order to get a meaningful result.

3. How is the integral of 1/x in complex variables evaluated?

The integral of 1/x in complex variables can be evaluated using the Cauchy Integral Formula, which states that for a function f(z) that is analytic in a simply connected domain D, the integral of f(z) along a closed contour C in D is equal to 2πi times the sum of the residues of f(z) at all singular points inside C. In the case of 1/x, the residue at z = 0 is 1, so the integral becomes 2πi.

4. What is the significance of the integral of 1/x in complex variables?

The integral of 1/x in complex variables has many applications in mathematics and physics. It is often used in complex analysis to solve problems related to contour integration and the behavior of functions in the complex plane. It is also used in the study of complex variables in quantum mechanics and electromagnetism.

5. Are there any real-world applications of the integral of 1/x in complex variables?

Yes, there are real-world applications of the integral of 1/x in complex variables. It is used in engineering and physics to solve problems related to electric fields, gravitational fields, and fluid flow. It also has applications in finance, such as in the Black-Scholes model for pricing options in the stock market.

Similar threads

Replies
14
Views
577
Replies
14
Views
1K
  • Calculus
Replies
3
Views
1K
Replies
3
Views
172
Replies
5
Views
1K
Replies
2
Views
136
Replies
6
Views
814
Replies
4
Views
1K
  • Calculus
Replies
29
Views
444
Replies
31
Views
704
Back
Top