Multivalued Complex Functions, Branch Cuts, and Riemann Surfaces
Bibliographic record
Abstract
After reading this appendix 1 you should be able to:• Understand concepts of complex valued multi-functions, branch cuts, and branch points • Represent complex valued multi-functions as the Riemann surface • View the Riemann surface corresponding to a multi-function as a uniquely defined object in multi-dimensional space• Understand how to preserve analyticity of a complex valued multifunction on a path crossing its branch cut. A.1 Multivalued Complex Functions, Branches, Branch Points, and Branch CutsWhat we commonly term as a complex valued function f (z) ∈ ℂ of complex variable z ∈ ℂ is a one-to-one mapping between complex values z ∈ ℂ and complex values 𝑤 ∈ ℂ defined as the solutions of equation 𝑤 = f (z), e.g.𝑤 = z 5 + 5. Mappingfrom complex-valued number z to complex-valued number 𝑤 is not such a ≪function≫ but rather a multifunction or multi-valued function as it produces two values of 𝑤 for each value of z, i.e. one value for n = 0 (referred to as the principle value of the square-root) and one for n = 1. 2 For a multifunction it is common to associate each of its distinct values with a branch.As such for two-valued function 𝑤 = √ z two branches for the two of its values, i.e. 𝑤 br.1 (z) and 𝑤 br.2 (z) corresponding to the cases n = 0 and n = 1, respectively, are introduced.Each of the branches of a multifunction is an analytic function everywhere on complex plane z except for a curve, at which the branches lose their analyticity.Such curve for a given multifunction is called the branch cut.The branch cut for a given multifunction is not uniquely defined and is subject to convention.For example, if for multifunction 𝑤 = √ z in Eq. (A.1) we define the argu-1 We recommend the reader to watch 'Imaginary Numbers are Real' videos by Welch Labs (https://youtu.be/ T647CGsuOVU?si=hRUA66Unc9lg86Fh)prior to reading this appendix to better understand presented concepts 2 Note other cases n = 2, 3, … are irrelevant since values of 𝑤 for them repeat those for n = 0 and n = 1.Theory and Computation of Electromagnetic Fields in Layered Media, First Edition.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.041 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".