Line Integral of a complex function

Click For Summary
SUMMARY

The discussion centers on solving the line integral \(\int \frac{dz}{8i + z^2}\) along the line \(y=x\) as \(x\) approaches infinity. The participant is restricted from using complex analysis tools such as Cauchy's theorem and seeks hints rather than direct answers. A suggestion is made to parameterize the line using \(z=re^{\pi i/4}\), which is a crucial step in simplifying the integral for evaluation.

PREREQUISITES
  • Understanding of line integrals in complex analysis
  • Familiarity with parameterization of curves
  • Basic knowledge of complex functions and their properties
  • Experience with limits and behavior of functions as variables approach infinity
NEXT STEPS
  • Study the parameterization of complex functions along specified paths
  • Learn about evaluating line integrals in the complex plane
  • Explore techniques for handling integrals without using Cauchy's theorem
  • Investigate the behavior of complex functions as they approach infinity
USEFUL FOR

Students and practitioners of complex analysis, mathematicians tackling line integrals, and anyone interested in advanced calculus techniques without relying on complex analysis theorems.

mateomy
Messages
305
Reaction score
0
I'm trying to solve this integral as x-> Infinity
[tex] \int \frac{dz}{8i + z^2}[/tex]

...on the y=x line, but I have no idea what I'm doing. The book I'm using is less than helpful in this regard. I'm not supposed to use any complex analysis tools (Cauchy, etc), but just solve it as a line integral. I'm not looking for an easy answer I would rather receive a hint. I've thought about expanding the function of [itex]z^2[/itex] and after doing so I'm still at a loss. Any advice would be great, thanks.
 
Physics news on Phys.org
mateomy said:
I'm trying to solve this integral as x-> Infinity
[tex] \int \frac{dz}{8i + z^2}[/tex]

...on the y=x line, but I have no idea what I'm doing. The book I'm using is less than helpful in this regard. I'm not supposed to use any complex analysis tools (Cauchy, etc), but just solve it as a line integral. I'm not looking for an easy answer I would rather receive a hint. I've thought about expanding the function of [itex]z^2[/itex] and after doing so I'm still at a loss. Any advice would be great, thanks.

You can parameterize the line y=x. Isn't that just [itex]z=re^{\pi i/4}[/itex]. Ok, turn the crank now.
 

Similar threads

Solve integral by finding Fourier series of complex function
  • · Replies 3 ·
Replies
3
Views
2K
Complex Integration Along Given Path
  • · Replies 3 ·
Replies
3
Views
2K
Can the contour integral of z⁷ be simplified using a parameterized expression?
  • · Replies 1 ·
Replies
1
Views
2K
Contour integration around a complex pole
  • · Replies 2 ·
Replies
2
Views
3K
Is f(z) =(1+z)/(1-z) a real function?
  • · Replies 17 ·
Replies
17
Views
4K
Show that the given function is continuous
  • · Replies 27 ·
Replies
27
Views
2K
Line integral of a scalar function about a quadrant
  • · Replies 4 ·
Replies
4
Views
2K
Arc length of vector function - the integral seems impossible
  • · Replies 2 ·
Replies
2
Views
2K
[Complex analysis] Contradiction in the definition of a branch
  • · Replies 8 ·
Replies
8
Views
3K
How Do You Solve a Complex Integral Using Cauchy-Goursat's Theorem?
  • · Replies 32 ·
2
Replies
32
Views
4K