Complex Analysis & Integral Transforms

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Abstractions about complex and generalized functions studied through analyticity, continuity, integration, transforms, singularities, and extension principles. They include harmonic and subharmonic structure, contour methods, kernels, residues, logarithms, regularization, and functional bounds.

29 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Absolute continuity — A strengthened continuity property that makes total function variation over sufficiently short disjoint intervals arbitrarily small and restores integration of the derivative.
• Baire one star function — A real function whose restriction is continuous on some nonempty relatively open piece of every perfect set.
• Bochner–Martinelli formula — An integral representation for continuously differentiable functions on domains in several complex variables using the Bochner–Martinelli kernel.
• Cauchy's integral formula — A boundary integral that reconstructs every value and derivative of a holomorphic function inside a contour.
• Contour integration — Integration of a complex function along an oriented path in the complex plane, with deformation and residues enabling evaluation.
• Dirac delta function — A distribution concentrated at one point whose action on a test function returns that function’s value at the point.
• Incomplete polylogarithm — A special function obtained by truncating the integral representation of the polylogarithm at a nonzero lower limit, also related to incomplete Fermi–Dirac and Bose–Einstein integrals.
• Indicator function (complex analysis) — A directional asymptotic function recording an entire function’s exponential growth rate.
• K-function — The special function extending the hyperfactorial to complex arguments through a functional equation involving powers and the gamma function.
• Koenigs function — A holomorphic change of coordinate that conjugates iteration near an attracting or repelling fixed point to multiplication by the fixed-point multiplier.
• Lacunary value — A complex number omitted from the image of a complex-valued function on its stated domain.
• Maximal function — A harmonic-analysis operator assigning each point the supremum of local averages of a function over neighborhoods containing or centered there.
• McShane integral — A gauge integral using free tagged partitions whose tags may lie outside their associated subintervals, yielding for real-valued functions exactly the Lebesgue-integrable class.
• Mellin transform — An integral transform mapping a function on positive scales to complex powers, making multiplicative scaling analogous to Fourier analysis of translation.
• Meromorphic function — A complex function holomorphic on a domain except at isolated poles, equivalently locally a quotient of holomorphic functions with nonzero denominator.
• Modulus of continuity — A nonnegative function bounding output variation in terms of input distance and tending to zero at zero, thereby quantifying uniform continuity.
• Monogenic function — In classical complex analysis, a function possessing a unique finite complex derivative at a point or throughout a stated set.
• Natural logarithm — The logarithm to base e, equivalently the inverse of the natural exponential and the signed area under one-over-x on the positive real line.
• Ohsawa–Takegoshi L2 extension theorem — A theorem extending square-integrable holomorphic functions from a complex submanifold with controlled norm.
• Phragmén–Lindelöf principle — A complex-analytic extension of the maximum-modulus principle that controls a holomorphic function on an unbounded domain using boundary bounds plus a growth restriction.
• Pluriharmonic function — A function on a complex manifold whose restriction to every complex line is harmonic, locally the real part of a holomorphic function in the real-valued case.
• Plurisubharmonic function — An upper-semicontinuous function on a complex domain whose restriction to every complex line is subharmonic.
• Positive-real function — A real-rational complex function analytic in the open right half-plane whose real part is nonnegative there, characterizing passive one-port impedances under standard conditions.
• Pseudoanalytic function — A complex-valued function satisfying a generalized first-order Cauchy–Riemann system determined by an admissible coefficient.
• Residue (complex analysis) — The coefficient of the inverse-linear term in a meromorphic function’s Laurent expansion at an isolated singularity.
• Singular function — A nonconstant continuous real function whose derivative exists and equals zero almost everywhere.
• Subharmonic function — An upper-semicontinuous function whose value at each point is no greater than the average over every sufficiently small surrounding sphere or ball.
• Unisolvent functions — A finite-dimensional function family for which interpolation values at any admissible set of distinct nodes determine one and only one function, equivalently yielding a nonsingular evaluation matrix.
• Zeta function regularization — An analytic-continuation method that assigns finite determinants, products, or sums to divergent spectral expressions through an associated zeta function.