Displaying similar documents to “Complex Integral”

Fundamental Group of n-sphere for n ≥ 2

Marco Riccardi, Artur Korniłowicz (2012)

Formalized Mathematics

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Triviality of fundamental groups of spheres of dimension greater than 1 is proven, [17]

More on the Continuity of Real Functions

Keiko Narita, Artur Kornilowicz, Yasunari Shidama (2011)

Formalized Mathematics

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In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.

Higher-Order Partial Differentiation

Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

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In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).

The Cauchy-Riemann Differential Equations of Complex Functions

Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura, Chanapat Pacharapokin (2009)

Formalized Mathematics

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In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.

Riemann Integral of Functions from R into n -dimensional Real Normed Space

Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama (2012)

Formalized Mathematics

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In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].

Lebesgue's Convergence Theorem of Complex-Valued Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2009)

Formalized Mathematics

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In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences...

Partial Differentiation of Real Binary Functions

Bing Xie, Xiquan Liang, Hongwei Li (2008)

Formalized Mathematics

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In this article, we define two single-variable functions SVF1 and SVF2, then discuss partial differentiation of real binary functions by dint of one variable function SVF1 and SVF2. The main properties of partial differentiation are shown [7].MML identifier: PDIFF 2, version: 7.9.03 4.104.1021

More on Continuous Functions on Normed Linear Spaces

Hiroyuki Okazaki, Noboru Endou, Yasunari Shidama (2011)

Formalized Mathematics

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In this article we formalize the definition and some facts about continuous functions from R into normed linear spaces [14].

Sorting by Exchanging

Grzegorz Bancerek (2011)

Formalized Mathematics

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We show that exchanging of pairs in an array which are in incorrect order leads to sorted array. It justifies correctness of Bubble Sort, Insertion Sort, and Quicksort.