In thermodynamics, characteristic function, aka "state functions", is a mathematical function that characterizes of the state of a given body.

Overview
In 1833, William Hamilton, in his “Problem of Three Bodies by my Characteristic Function”, penned in his Note Book 29, as found in his collected works (1940), built on the work of Joseph Lagrange and Pierre Laplace, to theorize about the the symbol V which he referred to as a “characteristic function”, defined as follows: [3]

Hamilton characteristic function

wherein m is the mass of three or four bodies, e.g. the sun and two planets, v is the velocity of the mass, mv² is the kinetic energy, or "living force" as it was called then, and t or dt was the "time element".

In 1869, French engineer Francois Massieu, in his “On the Various Functions Characteristic of Fluids”, introduced the concept of the existence of characteristic functions, in what seems to be the modern chemical thermodynamic sense of things. [1]

In 1876, American engineer Willard Gibbs summarized Massieu’s views as such: [2]

“Massieu has shown how all the properties of a fluid ‘which are considered in thermodynamics’ may be deduced from a single function, which he calls a characteristic function of the fluid considered; he introduces two different functions of this kind, vis, a function of the temperature and volume, which he denotes by Ψ, and a function of the temperature and pressure, which he denotes by Ψ’; in both cases he considers a constant quantity (one kilogram) of the fluid, which is regarded as invariable in composition.”

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Main functions
In terms of modern classification, there are generally considered to be five main characteristic functions::

Characteristic function:
Energy
U
Entropy
S
Enthalpy
U + PV
Helmholtz free energy
U – TS
Gibbs free energy
U + PV – TS

Description (synonyms):

(transformation content)
(heat content)

(constant: V,P free energy)
(work function)
(Helmholtz work function
)
(Helmholtz function)

(constant: T,P free energy)
(available energy)
(useful energy)

(Gibbs function)


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See also
‚óŹ Characteristic function notation table

References
1. (a) Massieu, Francois. (1869). “Sur les Functions Caracteristiques des Divers Fluides et Sur la Theorie des Vapeurs (On the Various Functions Characteristic of Fluids and on the Theory of Vapors)”, Comptes Rendus, 69: 858-62, 1057-61.
(b) Massieu, Francois. (1876). Thermodynamique: Mêmoire sur les fonctions catactéristiques des divers fluides et sur la théorie des vapeurs. 92-pgs. Académie des Sciences de L'Institut National de France.
2. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances" (Massieu, pgs. 86, 358),Transactions of the Connecticut Academy,III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.
3. Hamilton, William. (1940). The Mathematical Papers of William Rowan Hamilton, Volume Two, Dynamics (editors: A.W. Conway and A.J. McConnell) (abs, pg. xiii) (§1: Problem of Three Bodies by my Characteristic Function, pgs. 1-102; §2: On a General Method in Dynamics, pgs. 103-61; §: Second Essay on a General Method in Dynamics, pgs. 162-211). Cambridge University Press.

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