Math 2.0

Functions of a complex variable Theorem

Arguments add under multiplication

Builds on Argument of a complex numberPolar formProduct of exponentials

Statement

Let zz and ww be complex numbers with z≠0z \neq 0 and w≠0w \neq 0. If θ\theta is an argument of zz and φ\varphi is an argument of ww, then θ+φ\theta + \varphi is an argument of zwz w. Thus arg⁡(zw)≡arg⁡z+arg⁡w(mod2π)\arg(z w) \equiv \arg z + \arg w \pmod{2 \pi}.

Remarks

Proof: write z=∣z∣eiθz = |z| e^{i \theta} and w=∣w∣eiφw = |w| e^{i \varphi}. Then zw=∣z∣∣w∣ei(θ+φ)z w = |z| |w| e^{i (\theta + \varphi)} by the product of exponentials. For the principal values, the equality can fail. For example, Arg⁡(−1)+Arg⁡(−1)=2π\operatorname{Arg}(-1) + \operatorname{Arg}(-1) = 2\pi, but Arg⁡((−1)(−1))=0\operatorname{Arg}((-1)(-1)) = 0.

Reading path

To understand this page from the ground up, read these 15 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 abovedefinitionEuler's formula, statement, no evidence levelEuler's formulastatementProduct of exponentials, statement, no evidence levelProduct of exponentia…statementReal 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…statementPolar form, statement, no evidence levelPolar formstatementArguments add under multiplication, statement, no evidence levelArguments add under m…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.
  • no evidence level
  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Euler's formula statement
  5. Product of exponentials statement
  6. Real and imaginary parts definition
  7. Supremum definition
  8. Completeness axiom statement
  9. Existence of square roots of nonnegative reals statement
  10. Square root definition
  11. Modulus of a complex number definition
  12. Argument of a complex number definition
  13. Principal value of the argument statement
  14. Polar form statement
  15. Arguments add under multiplication statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Argument_(complex_analysis), Identities
About this page
Tag01W6RKJP01
URIm2://01W6RKJP01
Kindstatement, theorem
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 01W6RKJP01.
UsesArgument of a complex number, Polar form, Product of exponentials

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