Sentence examples for integral theorem from inspiring English sources

Using Cauchy's integral theorem, the integration path l can be deformed to the steepest descent path Γ. Recalling that the argument θ may range over ( 0, π ) and that the integrand in (3.1) has a simple pole at k = 1, we divide our discussion into three cases, namely (i)  0 < θ < π / 2, (ii) θ = π / 2, and (iii) π 2 < θ < π.

Goursat was one of the leading analysts of his time, and his detailed analysis of Augustin Cauchy's work led to the Cauchy-Goursat theorem, which eliminated the redundant requirement of the derivative's continuity in Cauchy's integral theorem.

It is based on using Cauchy's integral theorem and solving an initial-value problem.

This reminds us to utilize again Stein's oscillatory integral theorem.

The numerical results are compared with the analytic results derived from the Kirchhoff–Helmholtz integral theorem.

A unified approach, originating from Cauchy integral theorem, is presented to derive boundary integral equations for two dimensional elasticity problems.

The main tools in the proof of the Sobolev inequalities for Riemannian manifolds are the Hadamard parametrix (cf. L. Hörmander, Acta Math.88 1968, 341 370, and "The Analysis of Linear Partial Differential Equations," Vol. III, Springer-Verlag, New York 1985) and oscillatory integral theorems of L. Carleson and P. Sjölin (Studia Math.44 1972, 287 299) and Stein Annn. Math. Stud.112 1986, 307 357).

They proved a maximal inequality in (L^{2}) for stochastic convolution integrals (Theorem 1).

Our results about the boundedness of (mu_{j}^{L}) and ([b,mu_{j}^{L}]) from (M_{p,varphi_{1}}^{alpha,V}) to (M_{p,varphi_{2}}^{alpha,V}) (Theorems 1.1 and 1.2) are based on the local estimates for the Marcinkiewicz integrals (Theorem 3.1) and their commutators (Theorem 4.1).

In the year 1960, Opial [1] established the following integral inequality: Theorem 1.1.

We present the following assertion about χ-estimates for a multivalued integral [[18], Theorem 4.2.3].