Math 2.0

Functions of a complex variable Proposition

Principal value of the argument

Builds on Argument of a complex number

Statement

Let zz be a complex number with z≠0z \neq 0. Then zz has exactly one argument θ\theta with −π<θ≤π-\pi < \theta \le \pi. This argument is the principal value of the argument, written Arg⁡z\operatorname{Arg} z.

Remarks

Proof: the arguments of zz form a set θ0+2πℤ\theta_0 + 2 \pi \mathbb{Z}, and each interval (−π,π](-\pi, \pi] of length 2π2\pi contains exactly one element of such a set. For example, Arg⁡(−1)=π\operatorname{Arg}(-1) = \pi, Arg⁡(−i)=−π/2\operatorname{Arg}(-i) = -\pi/2 and Arg⁡x=0\operatorname{Arg} x = 0 for real x>0x > 0. Some authors use the interval [0,2π)[0, 2\pi) instead.

Reading path

To understand this page from the ground up, read these 11 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 abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness 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 …definitionArgument of a complex number, definition, no evidence levelArgument of a complex…definitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statement
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.
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Completeness axiom statement
  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Argument of a complex number definition
  11. Principal value of the argument statement

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Argument_(complex_analysis), Principal value
About this page
Tag0SQSQP748R
URIm2://0SQSQP748R
Kindstatement, proposition
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 0SQSQP748R.
UsesArgument of a complex number

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