$\ker (\exp )=2\pi i\mathbb{Z}$, 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
• Distinct pure frequencies differing by a reciprocal-lattice shift have identical samples Counterexample
• 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
• Forgetting the compact-open topology destroys Pontryagin duality Counterexample
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• The exponential map has no finite proper-map degree Counterexample
• The exponential map is a holomorphic surjection C to C^× that is not an automorphism Counterexample
• The radix-two split fails for odd N Counterexample
• Continuous logarithms and continuous arguments along a contour Definition
• Full-rank lattices, covolume, and the dual lattice Definition
• Roots of a compact connected Lie group Definition
• The recursive radix-two fast Fourier transform Definition
• The unitary discrete Fourier transform on ℤ/Nℤ Definition
• The unnormalised engineering DFT and its conversion to the unitary transform Definition
• 1=e^2π i does not imply 0=2π i: logarithms invert the exponential only modulo its kernel Example
• Annihilators of closed subgroups of Euclidean space Example
• Annulus and punctured disc have hyperbolic universal covers Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• Continuing the logarithm once around the unit circle adds 2 pi i Example
• Cyclic convolution on ℤ/4ℤ via the DFT Example
• Deficiencies of the exponential and sine Example
• Exponential omits two sphere values Example
• Four shared values do not force equality Example
• Haar normalisations on a finite abelian group and its dual Example
• Haar normalisations on the circle and the integers Example
• Nearest-grid and tail estimates in phase estimation Example
• Square and hexagonal lattice invariants Example
• The bidual map on the circle and the integers Example
• The exponential function omits exactly zero and shows little Picard is sharp Example
• The five characters of Z/5Z and their orthogonality Example
• The four-point radix-two FFT executed in full Example
• The function e^1/z omits zero and takes every nonzero value infinitely often near the origin Example
• The length-five all-ones vector through Bluestein Example
• The periodic Poisson circulant has one compatible zero mode Example
• The rank-one cotangent and its conic Example
• The unitary DFT for N=1 and N=2 Example
• A disc missing p carries a holomorphic logarithm of z-p Lemma
• A Quantitative Logarithm Near the Identity in SU(2) Lemma
• Circular-distance tail bound in phase estimation Lemma

…and 45 more results.

Dependency tree · two levels

25 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)