### Inverse Trigonometric Functions arcsec and arccosec

- FORMALIZED MATHEMATICS VOL. 16, NO. 2, PAGES 159–165, 2008
, 2008

"... This article describes definitions of inverse trigonometric functions arcsec and arccosec, as well as their main properties. ..."

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This article describes definitions of inverse trigonometric functions arcsec and arccosec, as well as their main properties.

### Integrability formulas. Part III

, 2010

"... In this article, we give several differentiation and integrability ..."

### 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 ..."

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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]. ..."

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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 Integrability Formulas of Some Functions, Orthogonal Polynomials and Norm Functions

"... Summary. In this article, we give several integrability formulas of some functions including the trigonometric function and the index function [3]. We also give the definitions of the orthogonal polynomial and norm function, and some of their important properties [19]. ..."

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Summary. In this article, we give several integrability formulas of some functions including the trigonometric function and the index function [3]. We also give the definitions of the orthogonal polynomial and norm function, and some of their important properties [19].

### Difference and Difference Quotient. Part II

, 2008

"... Summary. In this article, we give some important properties of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions [11]. ..."

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Summary. In this article, we give some important properties of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions [11].

### 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. ..."

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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.

### Basic Properties of Periodic Functions

"... Summary. In this article we present definitions, basic properties and some examples of periodic functions according to [4]. ..."

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Summary. In this article we present definitions, basic properties and some examples of periodic functions according to [4].

### Difference and Difference Quotient. Part IV

, 2011

"... Summary. In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions. ..."

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Summary. In this article, we give some important theorems of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions.