Riemann Integral of Functions R into C

Open access

Riemann Integral of Functions R into C

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].

[1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121-124, 1993.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[5] Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.

[6] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

[7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[9] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.

[10] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.

[11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.

[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.

[13] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.

[14] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.

[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.

[16] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.

[17] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.

[18] Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.

[19] Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.

[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

Formalized Mathematics

(a computer assisted approach)

Journal Information


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Target Group

researchers in the fields of formal methods and computer-checked mathematics

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 27 27 5
PDF Downloads 7 7 1