Displaying similar documents to “The Cauchy-Riemann Differential Equations of Complex Functions”

Riemann Integral of Functions from ℝ into Real Banach Space

Keiko Narita, Noboru Endou, Yasunari Shidama (2013)

Formalized Mathematics

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In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements of a real normed space and finite sequences of real numbers. In addition we proved some theorems...

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.

Complex Function Differentiability

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

Formalized Mathematics

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For a complex valued function defined on its domain in complex numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential complex functions are shown.

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).

Partial Differentiation of Real Ternary Functions

Takao Inoué, Bing Xie, Xiquan Liang (2010)

Formalized Mathematics

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In this article, we shall extend the result of [19] to discuss partial differentiation of real ternary functions (refer to [8] and [16] for partial differentiation).

The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

Keiko Narita, Noboru Endou, Yasunari Shidama (2013)

Formalized Mathematics

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In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the...

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].

The Differentiable Functions from R into R n

Keiko Narita, Artur Korniłowicz, Yasunari Shidama (2012)

Formalized Mathematics

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In control engineering, differentiable partial functions from R into Rn play a very important role. In this article, we formalized basic properties of such functions.

The C k Space

Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama (2013)

Formalized Mathematics

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In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].

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].

Differentiable Functions on Normed Linear Spaces

Yasunari Shidama (2012)

Formalized Mathematics

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In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant...