$\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$

Depends on

• $\exp(z+w)=\exp z\,\exp w$, and the complex exponential extends the real exponential
• Euler's formula: $\exp(i\theta)=\cos\theta+i\sin\theta$ for every real $\theta$
• Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
• Pythagorean and parity identities for all six trigonometric functions on their natural domains
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
• Quarter-turn values and shifts by pi/2 and pi
• Pi as twice the smallest positive zero of cosine

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
• Cartesian and polar forms of the Cauchy–Riemann equations agree away from the origin Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• The winding number is the increment of a continuous argument divided by 2π Corollary
• Trigonometric polynomials are uniformly dense in continuous functions on the torus Corollary
• A holomorphic function on an annulus can have a nonzero closed-contour integral 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
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• Irrational rotation is ergodic but not weakly mixing Counterexample
• The exponential map is a holomorphic surjection C to C^× that is not an automorphism Counterexample
• The family e^(nz) converges chordally to infinity on the right half-plane without being holomorphically normal there Counterexample
• The map (z₁,z₂)↦(e^z₁,z₂) has invertible complex Jacobian everywhere and is not injective Counterexample
• Characteristic function of a real random variable Definition
• Fourier coefficients and trigonometric polynomials on the torus Definition
• Fourier transform on complex L1 classes Definition
• Roots of a compact connected Lie group Definition
• The one-dimensional torus and its normalized Haar integral Definition
• 1=e^2π i does not imply 0=2π i: logarithms invert the exponential only modulo its kernel Example
• A continuous argument computed along a spiralling contour Example
• A deterministic integral construction of a Gaussian process Example
• Banach-Stone weighted composition isometries Example
• Cauchy estimates on a bidisc, computed and compared with the exact derivatives Example
• Independent sums via characteristic functions Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• Poisson kernel transform and Abel summability on the line Example
• Sinc-square integral from Plancherel Example
• The complex exponential satisfies the Cauchy–Riemann equations in Cartesian and polar form Example
• The logarithms of -1 are (2k+1)π i, k∈ℤ Example
• The unit circle traversed three times has index 3 at every interior point Example
• Transform of an interval indicator Example
• FALSE: boundary control alone gives the maximum principle on an unbounded domain False statement
• Basic properties of characteristic functions Lemma
• Characteristic function of a normal law Lemma
• Euclidean Gaussian transform with the 2π normalization Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• Finite sums of the sine harmonics Lemma
• Finite tori are compact Hausdorff spaces separated by characters Lemma
• Fourier uniqueness for continuous functions on the Euclidean torus Lemma

…and 27 more results.

Dependency tree · two levels

32 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)