Math 2.0

Functions of a complex variable Theorem

Polar form

Builds on Principal value of the argumentModulus of a complex numberEuler's formula

Statement

Let zz be a complex number with z≠0z \neq 0, let r=∣z∣r = |z| be its modulus and let θ=Arg⁡z\theta = \operatorname{Arg} z be its principal argument. Then z=r(cos⁡θ+isin⁡θ)=reiθ.z = r (\cos \theta + i \sin \theta) = r e^{i \theta}.

Remarks

The second equality is Euler's formula. The pair (r,θ)(r, \theta) gives the polar coordinates of the point zz in the complex plane. Polar form makes products easy: reiθ⋅seiφ=rs ei(θ+φ)r e^{i \theta} \cdot s e^{i \varphi} = r s \, e^{i (\theta + \varphi)}.

Reading path

To understand this page from the ground up, read these 13 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 formulastatementReal 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 formstatement
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. Euler's formula statement
  5. Real and imaginary parts definition
  6. Supremum definition
  7. Completeness axiom statement
  8. Existence of square roots of nonnegative reals statement
  9. Square root definition
  10. Modulus of a complex number definition
  11. Argument of a complex number definition
  12. Principal value of the argument statement
  13. Polar form statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Polar_coordinate_system, Complex numbers
  2. Web https://en.wikipedia.org/wiki/Complex_number, Polar form
About this page
Tag0BRXFNQ5JV
URIm2://0BRXFNQ5JV
Aliasespolar-form-of-a-complex-number
Kindstatement, theorem
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 0BRXFNQ5JV.
UsesPrincipal value of the argument, Modulus of a complex number, Euler's formula

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