Jacobian Elliptic Functions. Eric Harold Neville. Weierstrass Carl Gustav Jakob Jacobi Gauss

Jacobian Elliptic Functions

Eric Harold Neville

Clarendon Press Oxford

1944

Ciências Exatas

Livro em bom estado de conservação, capa dura original. Good Blue Cloth Boards. The Clarendon press (1944) Language: English. 1st edn. Tall 8vo. Original gilt lettered blue cloth . Pp. xiii + 331. very good conditions.

The Neville theta functions can be defined in terms of the theta functions as ... The Jacobi elliptic functions can be represented as ratios of the Neville theta functions;


The Weierstrass zeta and sigma functions are not strictly elliptic functions since they are not periodic.

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e.g. the equation of the pendulum—also see pendulum (mathematics)).

They also have useful analogies to the functions of trigonometry, as indicated by the matching notation sn for sin. The Jacobi elliptic functions occur more in practical problems than the Weierstrass elliptic functions.

They were introduced by Carl Gustav Jakob Jacobi, around 1830.


Elliptic integrals came first, invented by the Bernoullis, and were studied by Maclaurin, Euler and Lagrange in the 18th century, and later by Legendre, when there was great interest in evaluating the integrals that appeared in scientific applications, after it was realized that most integrals could not be evaluated in terms of the elementary functions. Later, the brilliant and ingenious Gauss conceived of inverting the functions defined by incomplete elliptic integrals, unlocking a treasure chest of analytical investigations using the new methods of complex variables.

Gauss had the admirable habit of not publishing his work, completely avoiding the conflicts and acrimony among the lesser investigators who were eager to have their work, however inconsequential, and their priority, however doubtful, recognized, a drive that has not disappeared and dominates modern science, creating a mountain of mediocrity. Gauss did not need to blow his own kazoo, however, and was not concerned about having things he discovered given other men's names.

Scientific entrepreneurs kept posted on what Gauss was doing, so they could snatch bits and pieces for their own benefit.

The invention of elliptic functions is shared with C. G. J. Jacobi and Abel, who published their investigations around 1827, though Gauss knew many of the results as early as 1809.

Carl Gustav Jacob Jacobi (1804-1851) was born in Potsdam, the son of a wealthy Jewish banker. He became a Christian, probably to avoid restrictions on holding certain university posts. His work on elliptic functions began when he was very young, and held his interest throughout his career.

He went to Königsberg in 1826, where Bessel and Neumann worked, becoming associate professor in 1827 and remaining there for 18 years. Although he developed diabetes, he eventually died young of influenza and smallpox.

He should not be confused with his elder brother Moritz Hermann von Jacobi (1801-1874), who went to St. Petersburg in 1837 and apparently assumed a "von" for distinction. He was the Jacobi who worked on electric motors, ran an electric boat on the Neva, developed Baron Pavel Schilling's telegraph, invented electrotyping, and enunciated Jacobi's Law of energy transfer (a maximum when source and load impedances are the same).

Only the Jacobian elliptic functions will be discussed here, which are the ones most closely related to the familiar three types of elliptic integrals. There are many more elliptic functions, for example the Weierstrassian, as well as the related theta functions, all of which are important in the theory, and which are explained in Whittaker and Watson. My principal purpose here is only to make the Jacobian elliptic functions more familiar to the reader. For the detailed theory, refer to Whittaker and Watson. First, we must mention some preliminaries that will help the understanding.

The reader is no doubt familiar with the concept of the indefinite integral, which is the inverse of differentiation. If F(x) = x2, then f(x) = dF/dx = 2x, or F(x) = ∫ 2xdx = ∫ f(x)dx. The definite integral is taken between limits, say a and b, and is defined so that ∫(a,b) f(x)dx = F(b) - F(a), where we have used a notation more convenient in HTML that keeps things on one line. Here, b is the upper limit and a is the lower limit.

The definition of the integral is actually in terms of the definite integral, with its geometric interpretation as the area under the curve y = f(x), while the indefinite integral is a generalization.

In the definite integral, the variable of integration is a dummy variable, in that which letter is used to denote it is immaterial. ∫ (a,b) f(t)dt is exactly the same as ∫ f(u)du or ∫ f(x)dx.

When we have such a dummy variable, it is best to make it different from any other variable appearing in the problem to avoid any confusion (though this is often not done).



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