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
• Holomorphic roots of a nonvanishing function on a disc 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
• The exponential function omits 0 and infinity as a meromorphic map on the plane Counterexample
• An exponential contour integral approximated by Riemann sums and evaluated by parametrization and a primitive Example
• Banach-Stone weighted composition isometries Example
• Componentwise holomorphy checked for an explicit map ℂ²→ℂ³ Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 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 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
• Finite simple analytic families and their exact endpoint norms Lemma
• Maximum principle on a closed strip for bounded holomorphic functions Lemma
• The unit-disc estimate for Weierstrass elementary factors Lemma
• Zero free entire function of exponential type is an exponential Lemma
• A bounded harmonic function near an isolated puncture extends harmonically Theorem
• A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
• A plane domain is homologically simply connected exactly when every harmonic function has a global conjugate Theorem
• A slit-plane root branch biholomorphically parametrizes a sector Theorem
• Equivalent characterisations of a homologically simply connected domain Theorem
• Hadamard three-lines theorem Theorem
• On a positively oriented circle about a, the integral of (z-a)ᵐ is zero for every integer m except -1, and is 2 pi i for m=-1 Theorem
• Power maps are biholomorphisms on sectors of width less than 2π/n Theorem
• Schottky's theorem Theorem
• The exponential is the inverse biholomorphism from the principal strip to the slit plane Theorem
• The principal logarithm is a biholomorphism from the slit plane to the principal strip Theorem

Dependency tree · two levels

43 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)