By Pierre Basieux
ISBN-10: 3499611198
ISBN-13: 9783499611193
Read or Download Die Architektur der Mathematik. Denken in Strukturen. PDF
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Extra resources for Die Architektur der Mathematik. Denken in Strukturen.
Example text
C are not zeros ofu(z). , c constitute all the zeros of f(z). This means that f(z) has only finitely many zeros in the entire complex plane. Then, the entire function We have the asymptotic equation ^ n^ Ax ^ ex ^ Ax Ay x ^ y ^ In (Ay) y 58 ENTIRE FUNCTIONS But it follows from the formula that we found above for y that y ^ 2/ZT: and, consequently, In (2Ann) 27111 Thus,* \n(2Ann) y^J-2*/i i / A \ 1 Lii : In (Ay) = In 2Am ' 2K,I , An 2 + -^ . I n VlAnn) v ~,4 27zn J — In (LM-//) 4rc 4--/j- | = ln(2i4i:/0 + l n | l ^ In(2Am)-f ^ . Therefore, we obtain for the roots of Eq. -|-^^]. where n assumes positive integral values. ) The reader can convince himself by means of a substitution that Eq. 26. In the preceding section, we considered an equation of the form f(z) = AP(z), where f(z) is a transcendental entire function, P(z) is a polynomial, and A is a complex constant. Let us, finally, consider the equation f(*) = Ag(z), (72) THE THEOREM OF ALGEBRA AND P I C A R D ' S L I T T L E THEOREM 53 where f(z) and ^(z)are distinct transcendental entire functions. For example, let us consider the equation tan z — A, which can be written in the form s'mz = A cos z, or the equation e2 = A sin z, etc.