Where does residue collide?

Cauchy introduced the residue (French résidu) in his 1826 Exercices de mathématiques, defining it as the coefficient that remains when a function is expanded near a point where it blows up. In modern treatments such as Ahlfors's Complex Analysis, the residue at a pole a is the coefficient of the 1/(z-a) term in the function's Laurent series, and the residue theorem makes the integral of the function around a closed path equal to the sum of the enclosed residues times a fixed factor. The referent is a single complex number attached to an isolated singularity, and it is unrelated to the leftover of an estate, a unit in a protein, or an element of human conduct.[1][2]

Unit

One complex number: the coefficient of the 1/(z-a) term in the Laurent series at the point a

Recovery means

Not applicable

Measured by

Computed as a limit taken at the pole, or recovered by contour integration through the residue theorem

When two amino acids join, a water molecule is removed and the part of each that stays in the chain is a residue. Fischer's 1906 Untersuchungen über Aminosäuren, Polypeptide und Proteïne established that proteins are built from amino acids linked by peptide bonds, and the amino-acid unit that remains in that chain is now called a residue. The referent is a monomer unit inside a polypeptide, counted along a sequence, and it shares nothing with a coefficient at a pole, the leftover of an estate, or a recurring element of social conduct.[3]

Unit

One amino-acid residue: a single monomer unit within a polypeptide chain

Recovery means

Not applicable

Measured by

Identified and counted by protein sequencing, mass spectrometry, and structure determination

Pareto named residues (Italian residui) in his 1916 Trattato di sociologia generale, translated as The Mind and Society in 1935, to mark the relatively constant elements of human conduct that remain once the shifting justifications people give for their actions are stripped away. For Pareto these residues express deep, stable sentiments and drive much non-logical behaviour, while the surface reasonings, which he called derivations, vary from case to case. The referent is a recurring element of social conduct inferred across many human actions, and it is not a coefficient at a pole, the leftover of an estate, or a unit in a protein.[4][5]

Unit

One residue: a recurring non-logical element that expresses an underlying sentiment

Recovery means

Not applicable

Measured by

Inferred by Pareto from patterns across large numbers of recorded human actions and beliefs