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Complex logarithms, the principal logarithm, and principal and multivalued complex powers
Definition
For with principal polar form , , define the principal logarithm The set of all complex logarithms of is For , define the principal power and the multivalued power respectively by Thus the first is one specified complex number and the second is a set of complex numbers; neither notation silently identifies them. The conventions and prerequisite facts used below are recorded in Every nonzero complex number has a unique polar form with and , The natural logarithm as the inverse of the exponential function, The complex exponential by its power series, is a field, every element is uniquely , and every nonzero element has inverse .
Depends on
- Every nonzero complex number has a unique polar form $r(\cos\theta+i\sin\theta)$ with $r>0$ and $-\pi<\theta\le\pi$
- The natural logarithm as the inverse of the exponential function
- The complex exponential by its power series
- $\mathbb C=\mathbb R[x]/(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)/(a^2+b^2)$
Used by
- The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
- ((-1)²)^1/2=1≠-1=(-1)^2(1/2) for principal complex powers Counterexample
- Continuous logarithms and continuous arguments along a contour Definition
- Standard semicircle, rectangle, keyhole, indentation, and sector contours Definition
- All values of iⁱ are the positive real numbers e^-π/2-2π k, k∈ℤ Example
- Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
- A keyhole contour sees the two boundary values of z^(alpha-1) Lemma
- Dictionary for holomorphic logarithm branches, the principal logarithm, and principal powers Remark
- All logarithms of z≠0 are Logz+2π i k, k∈ℤ Theorem
- Hadamard three-lines theorem Theorem
- There is no continuous logarithm on all of ℂ∖{0} Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)