Math 2.0

Functions of a complex variable Singularities and residues Theorem

Argument principle

Builds on Meromorphic functionOrder of a zeroOrder of a poleCircle integral

Statement

Let c∈ℂc \in \mathbb{C} and r>0r > 0. Let ff be a meromorphic function on an open set that contains the closed disc {z:∣z−c∣≤r}\{ z : |z - c| \le r \}. Suppose that ff has no zeros and no poles on the circle ∣z−c∣=r|z - c| = r. Let ZZ be the number of zeros of ff in the open disc ∣z−c∣<r|z - c| < r, each counted as many times as its order. Let PP be the number of poles of ff in the open disc, each counted as many times as its order. Then 12πi∮∣z−c∣=rf′(z)f(z) dz=Z−P,\frac{1}{2\pi i} \oint_{|z - c| = r} \frac{f'(z)}{f(z)} \, dz = Z - P, where the circle integral goes once around the circle counterclockwise.

Remarks

The function f′/ff'/f is the logarithmic derivative of ff. At a zero of order mm, it has a simple pole with residue mm; at a pole of order mm, it has a simple pole with residue −m-m. So the theorem follows from the residue theorem. The left side is also the winding number around 00 of the closed curve θ↦f(c+reiθ)\theta \mapsto f(c + r e^{i\theta}), θ∈[0,2π]\theta \in [0, 2\pi].

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionComplex exponential, definition, no evidence levelComplex exponentialdefinitionContour integral, definition, no evidence levelContour integraldefinitionPower series in the complex plane, definition, no evidence levelPower series in the c…definitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCircle integral, definition, no evidence levelCircle integraldefinitionDisc of convergence, definition, no evidence levelDisc of convergencedefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionCauchy's integral formula, statement, no evidence levelCauchy's integral for…statementIsolated singularity, definition, no evidence levelIsolated singularitydefinitionZero of a holomorphic function, definition, no evidence levelZero of a holomorphic…definitionPower series are holomorphic in the disc of convergence, statement, no evidence levelPower series are holo…statementPole, definition, no evidence levelPoledefinitionHolomorphic functions are analytic, statement, no evidence levelHolomorphic functions…statementMeromorphic function, definition, no evidence levelMeromorphic functiondefinitionHolomorphic functions are infinitely differentiable, statement, no evidence levelHolomorphic functions…statementOrder of a zero, definition, no evidence levelOrder of a zerodefinitionOrder of a pole, definition, no evidence levelOrder of a poledefinitionArgument principle, statement, no evidence levelArgument principlestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Complex exponential definition
  5. Contour integral definition
  6. Power series in the complex plane definition
  7. Real and imaginary parts definition
  8. Supremum definition
  9. Circle integral definition
  10. Disc of convergence definition
  11. Completeness axiom statement
  12. Existence of square roots of nonnegative reals statement
  13. Square root definition
  14. Modulus of a complex number definition
  15. Complex differentiability definition
  16. Holomorphic function definition
  17. Cauchy's integral formula statement
  18. Isolated singularity definition
  19. Zero of a holomorphic function definition
  20. Power series are holomorphic in the disc of convergence statement
  21. Pole definition
  22. Holomorphic functions are analytic statement
  23. Meromorphic function definition
  24. Holomorphic functions are infinitely differentiable statement
  25. Order of a zero definition
  26. Order of a pole definition
  27. Argument principle statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Argument_principle, Formulation
About this page
Tag0PN2BXRS8G
URIm2://0PN2BXRS8G
Aliasescauchys-argument-principle
Kindstatement, theorem
WritingW1 stated
MSC30C15
StatusThis copy of the site has no build data. Run m2 status 0PN2BXRS8G.
UsesMeromorphic function, Order of a zero, Order of a pole, Circle integral
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