Results

**1 - 7**of**7**### Integrability formulas. Part II

, 2010

"... In this article, we give several differentiation and integrability formulas of special and composite functions including trigonometric function, and polynomial function. ..."

Abstract
- Add to MetaCart

In this article, we give several differentiation and integrability formulas of special and composite functions including trigonometric function, and polynomial function.

### The Sum and Product of Finite Sequences of Complex Numbers

, 2010

"... Summary. This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11]. MML identifier: RVSUM 2, version: 7.11.07 4.156.1112 The notation and terminology used in this paper have been introduced in the ..."

Abstract
- Add to MetaCart

(Show Context)
Summary. This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11]. MML identifier: RVSUM 2, version: 7.11.07 4.156.1112 The notation and terminology used in this paper have been introduced in the following papers: [5], [7], [6], [4], [8], [13], [9], [2], [3], [15], [10], [12], and [14]. Auxiliary Theorems Let F be a complex-valued binary relation. Then rng F is a subset of C. Let D be a non empty set, let F be a function from C into D, and let F 1 be a complex-valued finite sequence. Note that F · F 1 is finite sequence-like. For simplicity, we adopt the following rules: i, j denote natural numbers, x, x 1 denote elements of C, c denotes a complex number, F , F 1 , F 2 denote complex-valued finite sequences, and R, R 1 denote i-element finite sequences of elements of C. The unary operation sqrcomplex on C is defined as follows: (Def. 1) For every c holds (sqrcomplex)(c) = c 2 . Next we state two propositions: Let us observe that the functor F 1 + F 2 is commutative. Let us consider i, R 1 , R 2 . Then R 1 + R 2 is an element of C i . The following propositions are true: Let us consider F . Then −F is a finite sequence of elements of C and it can be characterized by the condition: Let us consider i, R. Then −R is an element of C i . The following propositions are true: Let us consider F 1 , F 2 . Then F 1 − F 2 is a finite sequence of elements of C and it can be characterized by the condition: The following propositions are true: Let us consider F , c. We introduce c · F as a synonym of c F. The sum and product of finite sequences of . . . 109 Let us consider F , c. Then c · F is a finite sequence of elements of C and it can be characterized by the condition: One can prove the following four propositions: is a finite sequence of elements of C and it can be characterized by the condition: Let us note that the functor Next we state four propositions: Finite Sum of Finite Sequence of Complex Numbers One can prove the following propositions: 110 keiichi miyajima and takahiro kato The Product of Finite Sequences of Complex Numbers One can prove the following propositions: Modified Part of [1] We now state several propositions: (50) For every complex-valued finite sequence x holds len(−x) = len x. (51) For all complex-valued finite sequences x 1 , x 2 such that len x 1 = len x 2 holds len(x 1 + x 2 ) = len x 1 . (52) For all complex-valued finite sequences x 1 , x 2 such that len x 1 = len x 2 holds len(x 1 − x 2 ) = len x 1 . (53) For every real number a and for every complex-valued finite sequence x holds len(a · x) = len x. (54) For all complex-valued finite sequences x, y, z such that len x = len y = len z holds (x + y) References [1] Kanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of n-dimensional topological space. Formalized Mathematics, 11

### Several Integrability Formulas of Special Functions. Part II

"... Summary. In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function [3]. ..."

Abstract
- Add to MetaCart

(Show Context)
Summary. In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, the hyperbolic function and the polynomial function [3].

### Several Differentiation Formulas of Special Functions. Part VII

"... Summary. In this article, we prove a series of differentiation identities [2] involving the arctan and arccot functions and specific combinations of special functions including trigonometric and exponential functions. ..."

Abstract
- Add to MetaCart

(Show Context)
Summary. In this article, we prove a series of differentiation identities [2] involving the arctan and arccot functions and specific combinations of special functions including trigonometric and exponential functions.

### Integrability Formulas. Part I

, 2010

"... In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, and the polynomial function. ..."

Abstract
- Add to MetaCart

(Show Context)
In this article, we give several differentiation and integrability formulas of special and composite functions including the trigonometric function, and the polynomial function.

### Basic Properties of Even and Odd Functions

"... Summary. In this article we present definitions, basic properties and some ..."

Abstract
- Add to MetaCart

(Show Context)
Summary. In this article we present definitions, basic properties and some