Riemann Integral of Functions R into C

Miyajima, Keiichi; Kato, Takahiro; Shidama, Yasunari

doi:10.2478/v10037-010-0024-8

Abstract

In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the [19].

References

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121-124, 1993.
Search in Google Scholar
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
Search in Google Scholar
[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
Search in Google Scholar
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
Search in Google Scholar
[5] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
Search in Google Scholar
[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
Search in Google Scholar
[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
Search in Google Scholar
[8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
Search in Google Scholar
[9] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
Search in Google Scholar
[10] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.
Search in Google Scholar
[11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.
Search in Google Scholar
[12] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.
Search in Google Scholar
[13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.
Search in Google Scholar
[14] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
Search in Google Scholar
[15] Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rⁿ. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.10.2478/v10037-009-0021-y
Search in Google Scholar
[16] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
Search in Google Scholar
[17] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
Search in Google Scholar
[18] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.
Search in Google Scholar
[19] Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.
Search in Google Scholar
[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
Search in Google Scholar
[21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Search in Google Scholar

Riemann Integral of Functions R into C

Abstract

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