What are the branch points for \log (z^2 + 2z + 3)?

  • Context: Graduate 
  • Thread starter Thread starter Gh0stZA
  • Start date Start date
  • Tags Tags
    Branch Points
Click For Summary

Discussion Overview

The discussion revolves around identifying the branch points of the logarithmic function \log(z^2 + 2z + 3). Participants explore the calculation of these branch points, the implications of the quadratic formula, and the analytic properties of the logarithm at specific points.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests assistance in calculating the branch points and finding a branch of \log(z^2 + 2z + 3) that is analytic at -1.
  • Another participant notes that the branch point of \log(z) is at z=0 and suggests solving the equation z^2 + 2z + 3 = 0 to find the branch points.
  • A participant claims to have found the branch points as -1 ± i, but later acknowledges a mistake after comparing with Wolfram Alpha, which provides -1 ± √2 i as the correct roots.
  • One participant asserts that Wolfram Alpha's result is correct, questioning the method used to arrive at the initial answer of -1 ± i.
  • After correcting their mistake, the participant asks about substituting the values back into the logarithmic expression, suggesting they found \log(2 - √2) and \log(2 + √2).
  • Another participant responds that no further action is needed to identify the branch points, emphasizing that substituting the zeros back into the quadratic should yield zero.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correct branch points, with one participant initially providing an incorrect answer and later aligning with another's correction. The discussion remains unresolved as to the implications of the identified branch points.

Contextual Notes

There are limitations in the discussion regarding the assumptions made in the calculations and the dependence on the quadratic formula. The resolution of the branch points is contingent on correctly applying these mathematical principles.

Gh0stZA
Messages
25
Reaction score
0
Hi everyone,

Could someone please help me calculate the branch points?

Find a branch of \log (z^2 + 2z + 3) that is analytic at -1, and compute the derivative at -1.
 
Physics news on Phys.org
The "branch point" of ln(z) itself is z= 0. So you need to solve z^2+ 2z+ 3= 0.
 
HallsofIvy said:
The "branch point" of ln(z) itself is z= 0. So you need to solve z^2+ 2z+ 3= 0.

I did. My answer is -1 \pm i but Wolfram Alpha gives it as -1 \pm \sqrt{2} i
 
Well, since you don't say how you got -1\pm i, I can only say that Wolfram Alpha is correct.
 
I'm sorry, I made a stupid mistake with my quadratic formula. I now have the same answer as Wolfram.

So do I basically substitute the values back into the expression within the logarithm? In that case, I get \log(2-\sqrt{2}) and \log(2+\sqrt{2})
 
Gh0stZA said:
I'm sorry, I made a stupid mistake with my quadratic formula. I now have the same answer as Wolfram.

So do I basically substitute the values back into the expression within the logarithm? In that case, I get \log(2-\sqrt{2}) and \log(2+\sqrt{2})

No. You need do nothing more to identify the branch points and sides, if you back-substituted the zeros of that quadratic back into the quad, you should get zero.
 

Similar threads

Undergrad Expansion of a complex function around branch point
  • · Replies 3 ·
Replies
3
Views
3K
Graduate How to Find Critical Points of function f(x,y,z)
  • · Replies 9 ·
Replies
9
Views
3K
Graduate A problem about branch cut in contour integral
  • · Replies 3 ·
Replies
3
Views
4K
Graduate Integrating a function of which poles appear on the branch cut
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Evaluating a Contour Integral of log(z) | Explaining Branch Cuts
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Branching Point at z=0 in f(z)=z^(1/2)
  • · Replies 1 ·
Replies
1
Views
1K
Bernoulli equation and parallel pipe branch
  • · Replies 31 ·
2
Replies
31
Views
5K
Graduate Complex: Defining Sqr(-1) in Main Branch of logz
  • · Replies 2 ·
Replies
2
Views
2K
Graduate How do I accurately compute a definite integral with a branch point?
  • · Replies 17 ·
Replies
17
Views
2K