Theorems in Complex Analysis: Abel's Theorem, Akhiezer's Theorem, Analyticity of Holomorphic Functions, Analytic Fredholm Theorem, Area Theorem (Con

9781156145111: Theorems in Complex Analysis: Abel's Theorem, Akhiezer's Theorem, Analyticity of Holomorphic Functions, Analytic Fredholm Theorem, Area Theorem (Con
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 67. Chapters: Abel's theorem, Akhiezer's theorem, Analyticity of holomorphic functions, Analytic Fredholm theorem, Area theorem (conformal mapping), Argument principle, Behnke–Stein theorem, Bloch's theorem (complex variables), Bôcher's theorem, Bohr–Mollerup theorem, Borel–Carathéodory theorem, Branching theorem, Carathéodory's theorem (conformal mapping), Carathéodory kernel theorem, Carleson–Jacobs theorem, Carlson's theorem, Casorati–Weierstrass theorem, Cauchy's integral formula, Cauchy's integral theorem, Cauchy–Hadamard theorem, Classification of Fatou components, Complex conjugate root theorem, Corona theorem, Denjoy–Wolff theorem, De Branges's theorem, De Moivre's formula, Edge-of-the-wedge theorem, Euler's formula, Fatou's theorem, Fundamental theorem of algebra, Gauss–Lucas theorem, Grunsky's theorem, Hadamard three-circle theorem, Hadamard three-lines theorem, Hardy's theorem, Harnack's principle, Hartogs' extension theorem, Hartogs' theorem, Hartogs–Rosenthal theorem, Hurwitz's theorem (complex analysis), Identity theorem, Identity theorem for Riemann surfaces, Jensen's formula, Koebe quarter theorem, König's theorem (complex analysis), Lagrange inversion theorem, Lindelöf's theorem, Liouville's theorem (complex analysis), Littlewood subordination theorem, Looman–Menchoff theorem, Maximum modulus principle, Measurable Riemann mapping theorem, Mergelyan's theorem, Mittag-Leffler's theorem, Monodromy theorem, Montel's theorem, Morera's theorem, Nachbin's theorem, Oka coherence theorem, Open mapping theorem (complex analysis), Ostrowski–Hadamard gap theorem, Paley–Wiener theorem, Phragmén–Lindelöf principle, Picard theorem, Polynomial function theorems for zeros, Radó's theorem (Riemann surfaces), Residue theorem, Riemann–Roch theorem, Rouché's theorem, Routh–Hurwitz theorem, Runge's theorem, Schottky's theorem, Schwarz lemma, Schwarz reflection principle, Schwarz–Ahlfors–Pick theorem, Sokhotski–Plemelj theorem, Titchmarsh convolution theorem, Ushiki's theorem, Weierstrass factorization theorem, Weierstrass preparation theorem, Wirtinger's representation and projection theorem. Excerpt: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients. Peter Rothe, in his book Arithmetica Philosophica (published in 1608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds “unless the equation is incomplete”, by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes...

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