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Complex Numbers and Functions [PDF]
, 1985 One of the advantages of dealing with the real numbers instead of the rational numbers is that certain equations which do not have any solutions in the rational numbers have a solution in real numbers. For instance, x 2 = 2 is such an equation. However, we also know some equations having no solution in real numbers, for instance x 2 = −1, or x 2 = −2 ...
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Complex Numbers and Number Sequences
1972The complex variable z is written in the form $$z = x + iy = r{e^{i\vartheta }}(x,y,r\vartheta real,r \geqq 0,\vartheta taken\bmod 2\pi ).$$
Gabor Szegö, George Pólya
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On the Multiplication of Complex Numbers
The Mathematical Gazette, 1949In the development of the theory of complex numbers, it is important to give a definition of them dependent only upon real numbers. In the usual algebraic treatments the product is postulated in the formor obtained from a matrix representation. The object of the present note is to show by elementary methods that the assumption that the multiplication ...
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1983
Publisher Summary This chapter discusses complex numbers. A number of the form a + ib, where a and b are real numbers, is called a complex number. Real numbers can be regarded as complex numbers for which b is zero. Thus, there is need only to consider complex numbers because real numbers will be contained within the system of complex numbers.
B.D. Bunday, H. Mulholland
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Publisher Summary This chapter discusses complex numbers. A number of the form a + ib, where a and b are real numbers, is called a complex number. Real numbers can be regarded as complex numbers for which b is zero. Thus, there is need only to consider complex numbers because real numbers will be contained within the system of complex numbers.
B.D. Bunday, H. Mulholland
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Complex Numbers and Quaternions
2012A complex number z is a number of the form a + bi where a and b are real numbers and i 2 = − 1. Cardona, who was a sixteenth century Italian mathematician, introduced complex numbers, and he used them to solve cubic equations. The set of complex numbers is denoted by ℂ, and each complex number has two parts namely the real part Re(z) = a, and the ...
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2016
We introduced complex numbers in Sect. 1.8 of the first volume. There we just defined the numbers themselves, but did not go any further. In fact, since the introduction of complex numbers a number of centuries ago, the theory based on them has been substantially developed into an extended analysis of complex functions defined on the complex plane. The
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We introduced complex numbers in Sect. 1.8 of the first volume. There we just defined the numbers themselves, but did not go any further. In fact, since the introduction of complex numbers a number of centuries ago, the theory based on them has been substantially developed into an extended analysis of complex functions defined on the complex plane. The
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1987
Let R be an entire ring. By an ordering of R one means a subset P of R satisfying the following conditions: ORD 1. For every x ∈ R we have x ∈ P, or x = 0, or — x ∈ P, and these three possibilities are mutually exclusive. ORD 2. If x, y ∈ P then x + y ∈ P and xy ∈ P.
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Let R be an entire ring. By an ordering of R one means a subset P of R satisfying the following conditions: ORD 1. For every x ∈ R we have x ∈ P, or x = 0, or — x ∈ P, and these three possibilities are mutually exclusive. ORD 2. If x, y ∈ P then x + y ∈ P and xy ∈ P.
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2011
An important new feature with respect to real analysis is the introduction of the point at infinity, which leads to the compactification of \( {\mathbb{C}} \). These various aspects, and some others, such as Moebius maps, are considered in this chapter.
Daniel Alpay, Daniel Alpay
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An important new feature with respect to real analysis is the introduction of the point at infinity, which leads to the compactification of \( {\mathbb{C}} \). These various aspects, and some others, such as Moebius maps, are considered in this chapter.
Daniel Alpay, Daniel Alpay
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Radiation therapy‐associated toxicity: Etiology, management, and prevention
Ca-A Cancer Journal for Clinicians, 2021Kyle Wang
exaly