$\ker (\exp )=2\pi i\mathbb{Z}$, and $\exp z=\exp w$ exactly when $z-w\in 2\pi i\mathbb{Z}$

Depends on

• $\exp(x+iy)=e^x(\cos y+i\sin y)$, $|\exp(x+iy)|=e^x$, and $e^{i\pi}+1=0$
• The exponential function is strictly increasing
• The zero sets of sine and cosine and the least positive common period 2 pi
• $t\mapsto(\cos t,\sin t)$ is a bijection from $[0,2\pi)$ onto the real unit circle

Used by

• A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• The exponential map is a holomorphic surjection C to C^× that is not an automorphism Counterexample
• Continuous logarithms and continuous arguments along a contour Definition
• Roots of a compact connected Lie group Definition
• 1=e^2π i does not imply 0=2π i: logarithms invert the exponential only modulo its kernel Example
• Continuing the logarithm once around the unit circle adds 2 pi i Example
• The exponential function omits exactly zero and shows little Picard is sharp Example
• The function e^(1/z) omits zero and takes every nonzero value infinitely often near the origin Example
• A disc missing p carries a holomorphic logarithm of z-p Lemma
• A circle traversed k times has winding number k inside and 0 outside Theorem
• A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
• All logarithms of z≠0 are Logz+2π i k, k∈ℤ Theorem
• Different branches shift logarithms by 2π i k and complex powers by exponential factors Theorem
• Every contour missing a point admits a continuous logarithm, unique up to a constant in 2π iℤ Theorem
• The index of a cycle about a point off its trace is an integer Theorem
• The integral of dz/(z-p) along a contour is the increment of a continuous logarithm Theorem
• The n-th roots of a complex number and the n distinct roots of unity for every n≥1 Theorem
• The Riemann surface of the logarithm is the complex plane over the punctured plane via exp Theorem
• The winding number of a closed contour is an integer Theorem
• The zeros of complex sine are the integer multiples of pi, and the zeros of complex cosine are the odd half-integer multiples of pi Theorem
• There is no continuous logarithm on all of ℂ∖{0} Theorem

Dependency tree · two levels

22 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)