In the neighbourhood of a finite value x0 of x, which is not a singularity of any of the coefficients in the equation, these integrals are ordinary power-series in x - x0.
To every group of finite order which can be represented as an irreducible group of linear substitutions on n variables will correspond a class of irreducible linear differential equations of the nth order whose integrals are algebraic.
Another application of groups of finite order is to the theory of linear differential equations whose integrals are algebraic functions.
When the closed path contains no singular point of the coefficients of the differential equation, the new set of integrals is identical with the original set.
The complete determination of the class of linear differential equations of the second order with all their integrals algebraic, whose group has the greatest possible order, viz.
The integrals are the integral duties of Christianity.
It is needful therefore to know the essentials of ordination, from the integrals and accidentals.
It is not ignorance or error about the mere integrals of christianity, which maketh them no christians who hold the essentials, that is, the baptismal covenant.
The following sketch of the proof of the existence of these principal integrals for the case n = 2 will show the character of more general investigations.
Conversely, as such a chain connectivity T can be taken out from the elements of any given integral all possible integrals are obtainable in this way.
Now in the integrals represented by G and H every element diminishes as V increases from zero.
We may then attempt to investigate, in general, in what cases the independent variable x of a hypergeometric equation is a single-valued function of the ratio s of two independent integrals of the equation.
We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y.
Conversely the existence of these principal integrals [omega]1, .
The integralsare then properly functions of the direction in which the light is to be estimated.
Investigation soon showed that certain integrals expressing relations between the motions not only of three but of any number of bodies could be found.
The third volume (1816) contains the very elaborate and now well-known tables of the elliptic integrals which were calculated by Legendre himself, with an account of the mode of their construction.
The remainder of the first volume relates to the Eulerian integrals and to quadratures.
Elliptic integral, one of an important class of integrals, occurring in the higher mathematics; -- so called because one of the integralsexpresses the length of an arc of an ellipse.
Eulerian integrals, certain definite integralswhose properties were first investigated by Euler.
When no external force acts, the case which we shall consider, there are three integrals of the equations of motion (i.
But one of them is simply the integral area between the curve and the datum line corresponding to the mean water level, and the others are the integrals of quantities of the type y sin nx.
All such definite integrals as the above are said to be "improper.
When m is indefinitely increased, the two last integrals have the limit zero in virtue of lemma (2).
The integrals expressing the area generated by QT have to be expanded in a series.
Quite different and simpler constructions can be used, if the integrals determining An and Bn be integrated by parts.
These integrals are obtained by the addition of ordinates, and therefore by an approximate method.
Definite integralscan sometimes be evaluated when the limits of integration are some particular numbers, although the corresponding indefinite integrals cannot be found.
The integrals are to be understood as line integrals taken along the curve.
A few additional results in relation to lineintegrals and multiple integrals are set down here.
For the calculation of definite integrals we have the following methods:-- (i.
New functions can be introduced also by means of the definite integrals of functions of two or more variables with respect to one of the variables, the limits of integration being fixed.
We have two theorems of transformation connecting volumeintegrals with surface integrals and surface integrals with line integrals.
Conversely, indefinite integrals are indeterminate to the extent of an additive constant.
The following are among the classes of functions whose integrals involve the elementary functions only: (i.
In contradistinction to multiple integrals the ordinary integral of a function of one variable with respect to that variable is called a "simple integral.
When the same integralsare taken between limits which correspond to two points of the curve, in the sense of line integrals along the arc between the points, they represent the area bounded by the arc and the terminal radii vectores.
Integrals such as surface and volume integrals are usually called "multiple integrals.
Since all simple integrals can be represented as areas such rules are applicable to approximate integration in general.
The difference of the two integrals will therefore be very small, and the probability will be very nearly 1/2.
We should even be puzzled to say which among all our integrals should retain the name of energy.
This circumstance must occur so much the oftener (the number of known integrals being still very small), seeing that these successive integrals are generally very different functions from the derivatives which have produced them.
Jacobi on elliptical functions have truly enlarged the field of analysis; and the same is true of some definite integrals obtained by M.
However, we know scarcely any thing in relation to irrational functions, the integrals of which have been obtained only in extremely limited cases, and particularly by rendering them rational.
This ingenious relation deserves to be noticed for another reason, as having suggested the first idea of that transformation of integrals yet unknown, which has lately received a greater extension, and of which M.
When the limits of the proposed integrals are to be fixed, their variations being zero, there is no occasion for noticing them.
The expression, always possible, of integrals in infinite series, may at first be viewed as a happy general means of compensating for the extreme imperfection of the integral calculus.
When a linear equation of the m^{th} order contains no term independent of the unknown function and its derivatives, the sum of any number whatever of particular integrals multiplied by arbitrary constants is also an integral.
Beauty, I hear you ask, do not the Graces flee where integrals stretch forth their necks?
The principal advantage arising from the use of hyperbolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions.
Columbia Mountain, promoted by the mergerers of Goldfield Consolidated, but excluded from the merger because not contiguous to the other integrals and because it had no ore, had been ballooned to $1.
I merely analyzed your tissues, surveyed the food-supplies you carried, discovered your individual preferences, and set up the necessary integralsin the mechanism.
The integrals render impossible any unwholesome, unpleasant, or unbalanced combination of any nature, and everything else is left to the mechanism, which operates upon pure chance.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "integrals" in a variety of sentences. We hope that you will now be able to make sentences using this word.
