The complex exponential is entire and its complex derivative is itself

Depends on

• Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set
• $\exp(x+iy)=e^x(\cos y+i\sin y)$, $|\exp(x+iy)|=e^x$, and $e^{i\pi}+1=0$
• The exponential function is smooth and $(\exp)'=\exp$
• The derivatives of sine and cosine are cosine and minus sine
• A function differentiable at $c$ is continuous at $c$
• Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\bar z}f=0$, or with the Cauchy–Riemann equations

Used by

• A holomorphic logarithm is a primitive of the logarithmic derivative Corollary
• A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order Corollary
• A plane harmonic function bounded above or below is constant Corollary
• Contraction Hille-Yosida theorem Corollary
• Holomorphic roots of a nonvanishing function on a disc Corollary
• The spherical complementary series converge to the trivial representation Corollary
• A holomorphic function on an annulus can have a nonzero closed-contour integral Counterexample
• e^1/z has an essential singularity at 0 and omits the value 0 Counterexample
• log|z| has no global harmonic conjugate on C{0 Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• The exponential function omits 0 and infinity as a meromorphic map on the plane Counterexample
• The exponential map has no finite proper-map degree Counterexample
• Inner, singular inner and outer functions Definition
• Spectral measures and the known-mean periodogram Definition
• The complex-time heat kernel on a proper sector Definition
• A compactly supported contrast with volume-operator norm greater than two Example
• A fixed partial-wave cutoff misses the forward high-frequency plane wave Example
• A half-circle configuration loop traces an elementary half twist Example
• A pure two-strand full twist as an ordered loop Example
• An exponential contour integral approximated by Riemann sums and evaluated by parametrization and a primitive Example
• Banach-Stone weighted composition isometries Example
• Characteristics of a monomial, an exponential and a tangent Example
• Componentwise holomorphy checked for an explicit map ℂ²→ℂ³ Example
• Deficiencies of the exponential and sine Example
• Exponential omits two sphere values Example
• Four shared values do not force equality Example
• Half-space Poisson extension of a plane wave Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• Shannon reconstruction of a sinc function Example
• The circle integral of eᶻ/(z-1) over |z|=2 is 2π i e Example
• The complex exponential satisfies the Cauchy–Riemann equations in Cartesian and polar form Example
• The exponential function omits exactly zero and shows little Picard is sharp Example
• The Gaussian integral on the line with a certified sinh-sinh truncation tail Example
• The polydisc boundary is not a smooth hypersurface, so the Szegő definition does not apply Example
• The regular representation of the real line as a multiplicity-one integral of characters Example
• The residue of eᶻ/z³ from the pole-derivative formula Example
• FALSE: boundary control alone gives the maximum principle on an unbounded domain False statement
• FALSE: every entire function with an antiderivative is a polynomial False statement
• FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy False statement
• A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma

…and 57 more results.

Dependency tree · two levels

45 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, Guide to Cultivating Complex Analysis, Exercise 2.1.4 (standard reference, not scraped)
• J. Orloff, MIT 18.04 Topic 2, Example 2.11 (standard reference, not scraped)