scholarly journals XXXVI. On the comparison of transcendents, with certain applications to the theory of definite integrals

1857 ◽  
Vol 147 ◽  
pp. 745-803 ◽  
Keyword(s):  
Algebraic Equation ◽  
Fundamental Theorem ◽  
Algebraic Function ◽  
Definite Integrals ◽  
Differential Coefficient ◽  
Finite Expression

1. The following objects are contemplated in this paper:— 1st. The demonstration of a fundamental theorem for the summation of integrals whose limits are determined by the roots of an algebraic equation. 2ndly. The application of that theorem to the problem of the comparison of algebraic transcendents. The immediate object of this application will in each case be the finite expression of the value of the sum of a series of integrals, Ʃ∫X dx , the differential coefficient, X, being an algebraic function, and the values of x at the limits being determined by the roots of an algebraic equation.

scholarly journals III. On the comparison of transcendents, with certain applications to the theory of definite integrals

1857 ◽  
Vol 8 ◽  
pp. 461-463 ◽  
Keyword(s):  
Algebraic Equation ◽  
Fundamental Theorem ◽  
Definite Integrals

The following objects are contemplated in this paper:— 1st. The demonstration of a fundamental theorem for the summation of integrals whose limits are determined by the roots of an algebraic equation. 2ndly. The application of that theorem to the comparison of algebraical transcendents.

On some definite integrals and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics

1903 ◽  
Vol 71 (467-476) ◽  
pp. 97-101 ◽  
Keyword(s):  
Spherical Harmonics ◽  
Fundamental Theorem ◽  
New Method ◽  
Definite Integrals

The expansion of a function f(θ) of an angle θ varying between 0 and π in terms of a series proceeding by the sines of the multiples of θ depends on the fundamental theorem, ∫ π 0 sin pθ sin qθ dθ = 0, where p and q are integer numbers not equal to each other.

Some Applications of Elliptic Integrals of First Kind

2013 ◽  
Vol 65 (1) ◽  
Author(s):  
Yasamin Barakat ◽  
Nor Haniza Sarmin
Keyword(s):  
Algebraic Function ◽  
High Accuracy ◽  
Elliptic Integrals ◽  
Definite Integrals ◽  
Engineering Mathematics ◽  
Optimal Values ◽  
Complete Elliptic Integrals

One of the most important applications of elliptic integrals in engineering mathematics is their usage to solve integrals of the form  (Eq. 1), where  is a rational algebraic function and  is a polynomial of degree  with no repeated roots. Nowadays, incomplete and complete elliptic integrals of first kind are estimated with high accuracy using advanced calculators.  In this paper, several techniques are discussed to show how definite integrals of the form (Eq. 1) can be converted to elliptic integrals of the first kind, and hence be estimated for optimal values. Indeed, related examples are provided in each step to help clarification.

XXVIII.—On General Differentiation. Part I

1840 ◽  
Vol 14 (2) ◽  
pp. 567-603 ◽  
Author(s):  
P. Kelland
Keyword(s):  
Exponential Function ◽  
Definite Integrals ◽  
Differential Coefficient ◽  
The Subject

We owe to Leibnitz the first suggestion of Differentiation, with fractional and negative indices, but no definite notion of the theory was attained until Euler expounded it in the Petersburgh Commentaries for 1731. Still Euler wrote only a few pages on the subject, so that the theory could scarcely be said to have come into existence, until Laplace, in his Théorie des Probabilités, and Fourier, in his Théorie de la Propagation de la Chaleur, shewed how general differential coefficients might be deduced by means of definite integrals, provided we assume or prove, by means of some elementary definition, that the differential coefficient of a circular or of an exponential function has a certain form. The formula given by M. Fourier is a very simple one; and our astonishment is great, when we reflect on the time which elapsed from its announcement to the first application that was made of it. This took place in 1832, in a memoir by M. Liouville, entitled Questions of Geometry and Mechanics resolved by a new analysis, which memóir is followed by two others on the more immediate theory of the analysis itself. Although M. Liouville regards his analysis as a new invention, we have no doubt that the idea is due to Fourier; but still to M. Liouville belongs the honour of moulding it into a shape capable of being made use of in the solution of problems.

scholarly journals Numerical Solution of Boole’s Rule in Numerical Integration by Using General Quadrature Formula

2012 ◽  
Vol 2 ◽  
pp. 1-4
Author(s):  
P.V. Ubale
Keyword(s):  
Numerical Solution ◽  
Numerical Integration ◽  
Quadrature Formula ◽  
Fundamental Theorem ◽  
Quadrature Method ◽  
Fundamental Theorem Of Calculus ◽  
Definite Integrals

We have seen that definite integrals arise in many different areas and that the fundamental theorem of calculus is a powerful tool for evaluating definite integrals. This paper describes classical quadrature method for the numerical solution of Boole’s rule in numerical integration.

Cosby's Rule: Connecting Summation Operations, Definite Integrals, and the Fundamental Theorem of Calculus

2010 ◽  
Vol 104 (1) ◽  
pp. 45-49
Author(s):  
Margaret H. Sloan
Keyword(s):  
Fundamental Theorem ◽  
Small School ◽  
Fundamental Theorem Of Calculus ◽  
Definite Integrals ◽  
Ap Calculus

What happened when a student in an AP Calculus class in a small school in rural Georgia discovered a way to bypass some laborious computations.

Some Mellin-type Definite Integrals Involving Sine and Cosine Functions

2019 ◽  
Vol 10 (1) ◽  
pp. 222-237
Author(s):  
M. I. Qureshi ◽  
Kaleem A. Quraishi ◽  
Dilshad Ahamad
Keyword(s):  
Definite Integrals ◽  
Sine And Cosine Functions ◽  
Cosine Functions
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