Implicit Functions


Also found in: Dictionary.
Related to Implicit Functions: implicit differentiation

Implicit Functions

 

functions defined by relations between independent variables that have not been explicitly solved for the latter; such relations are one of the methods of defining functions. For example, the relation

x2 + y2 − 1 = 0

defines an implicit function

y = y (x)

and the relations

x = ρ cos φ sin φ

x = ρ sin φ sin θ, z = ρ cos θ

define implicit functions

ρ = ρ(x, y, z) φ = φ(x, y, z) θ = θ(x, y, z)

In the simplest cases, the relations defining implicit functions may be solved in terms of elementary functions, that is, elementary functions may be found that satisfy these relations. Thus, in the first example above we have

and in the second,

However, in general, such elementary functions cannot be found. Implicit functions may be single-valued or many-valued. Not every relation (or system of relations) between variables defines an implicit function. Thus, if the values of the variables are restricted to real numbers, then the relation x2 + y2 + 1 = 0 does not define an implicit function, since the function exy = 0 is not satisfied by a single pair of real or complex numbers ξ and y; the relation e”y = 0 is not satisfied by a single pair of real or complex numbers ξ and y. The existence theorem for implicit functions in its simplest form asserts that if a function F (x, y) vanishes for a pair of values x = x0, and y = y0 [F (x0, y0 ≠ 0] and is differentiable in a neighborhood of the point (x0, y0 and, moreover, Fx(x, y) and Fy(x, y) are continuous in this neighborhood and Fy(x0, y0) ≠ 0, then in a sufficiently small neighborhood of the point x0 there exists a unique single-valued continuous function y = y (x) that satisfies the relation F (x, y) = 0 and takes on the value y0 for x = x0; also, y’(x) = —Fx(x, y)/Fy(x, y).

Power series are often used for the approximate calculation of values of an implicit function near x0, where the function’s value y0 is known. Thus, if F (x, y) is an analytic function—that is, it may be expanded as a convergent double power series in the neighborhood of the point (x0, y0)]—and if Fy(x0, y0) ≠ 0, then the implicit function defined by the relation F (x, y) = 0 may be obtained in the form of a power series

which is convergent in a certain neighborhood of the point x = x0 The coefficients Ck, k = 1, 2, . . . , may be found either by substituting this series in the relation F (x, y) = 0 or by successively differentiating this relation with respect to x. For example, if the implicit function is given by the relation

y5 + xy − 1 = 0, x0 = 0, y0 = 1

then

and

from which

If the relation F (x, y) = 0 may be put in the form y = a + xφ(y), where φ(y) is an analytic function, then the implicit function that is defined by this relation and that assumes the value a at x = 0 can be expanded in a Lagrange series

which is convergent in a certain neighborhood of the point x = 0. For example, the relation y = a + x sin y (Kepler’s equation) vields

In the general case, the values of an implicit function may be calculated by the method of successive approximations.

REFERENCES

Smirnov, V. I. Kurs vysshei matematiki, vol. 1, 22nd ed. Moscow, 1967. Vol. 3, part 2, 8th ed. Moscow, 1969.
Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 7th ed., vol. 1. Moscow, 1969.
Kudriavtsev, L. D. Matematicheskii analiz, vol. 2. Moscow, 1970.
References in periodicals archive ?
according to the known theorem of implicit functions (Litvin et al.
At solution of the one-objective sub-problems of complex optimisation with minimisation of some function ( ) F Y , and at continuous change of variables, the main roles are played by the first main clause of GMRG about application of the theory of implicit functions, i.
If the improvement of the solution for an one-criterion sub-problem in the complex optimisation, using special mathematical tools for acceleration of the computing process GMRG, is called complex optimal correction of ESS of EPS, the improvement of the solution of multi-objective sub-problems of the complex optimisation using special mathematical tools for acceleration of the computing process [application instead of the gradient of the one implicit function the array of the multi-objective descent or rise (dependent on the minimisation or maximisation of the partners objective functions)], is called complex multiobjective or Pareto-optimal correction of ESS of IPS.
i](z) will change now into the implicit functions of the vector x: [F.
Namely in the complex optimal control theory of big power systems (PS) and interconnected power systems (IPS), this approach has been implemented most widely and efficiently for solving hierarchic sub-problems of optimization within the framework of the so-called Generalized Reduced Gradient Method (GRGM) where, in addition to the implicit functions theory, a rational choice of a basis is used (i.