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Several differentiation formulas of special functions
 Part V. Formalized Mathematics
"... Summary. In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions. ..."
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Summary. In this article, we give several differentiation formulas of special and composite functions including trigonometric, polynomial and logarithmic functions.
Inverse Trigonometric Functions arctan and arccot
 FORMALIZED MATHEMATICS VOL. 16, NO. 2, PAGES 147–158, 2008
, 2008
"... This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot. ..."
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Cited by 7 (3 self)
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This article describes definitions of inverse trigonometric functions arctan, arccot and their main properties, as well as several differentiation formulas of arctan and arccot.
Determinant of Some Matrices of Field Elements
, 2006
"... Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matr ..."
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Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.
Properties of First and Second Order Cutting of Binary Relations
, 2005
"... This paper introduces some notions concerning binary relations according to [9]. It is also an attempt to complement the knowledge contained in the Mizar Mathematical Library regarding binary relations. We define here an image and inverse image of element of set A under binary relation of two sets ..."
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This paper introduces some notions concerning binary relations according to [9]. It is also an attempt to complement the knowledge contained in the Mizar Mathematical Library regarding binary relations. We define here an image and inverse image of element of set A under binary relation of two sets A, B as image and inverse image of singleton of the element under this relation, respectively. Next, we define “The First Order Cutting Relation of two sets A, B under a subset of the set A ” as the union of images of elements of this subset under the relation. We also define “The Second Order Cutting Subset of the Cartesian Product of two sets A, B under a subset of the set A” as an intersection of images of elements of this subset under the subset of the Cartesian Product. The paper also defines first and second projection of binary relations. The main goal of the article is to prove properties and collocations of definitions introduced in this paper.
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. ..."
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In this article, we give several differentiation and integrability formulas of special and composite functions including trigonometric function, and polynomial function.
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 complexvalued 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 complexvalued finite sequence. Note that F · F 1 is finite sequencelike. 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 complexvalued finite sequences, and R, R 1 denote ielement 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 complexvalued finite sequence x holds len(−x) = len x. (51) For all complexvalued 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 complexvalued 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 complexvalued finite sequence x holds len(a · x) = len x. (54) For all complexvalued 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 ndimensional topological space. Formalized Mathematics, 11
Some Differentiable Formulas of Special Functions
"... Summary. This article contains some differentiable formulas of special functions. ..."
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Summary. This article contains some differentiable formulas of special functions.
Several Differentiable Formulas of Special Functions
"... Summary. In this article, we give several differentiable formulas of special functions. There are some specific composite functions consisting of rational functions, irrational functions, trigonometric functions, exponential functions or logarithmic functions. ..."
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Summary. In this article, we give several differentiable formulas of special functions. There are some specific composite functions consisting of rational functions, irrational functions, trigonometric functions, exponential functions or logarithmic functions.
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].