Elements of the Theory of Functions of a Complex Variable With Especial Referenc
Elements of the Theory of Functions of a Complex Variable With Especial Referenc
H Heinrich Durge
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E. , by letting the point z approach the origin along the spiral of Archimedes which is explicitly determined by the value a, and which is tangent to the y-axis at the origin. f*OS ih With this relation existing between \j/ and r } " now be- T comes infinite as r decreases indefinitely, and therefore -the tangent to this curve is capable of assuming every value. But if we denote by a the definite arc contained between J 2 7T k and -, the tangent of which has the value, so that the 2 h arbitrari...ly assumed values h and k can be replaced by the equally arbitrary quantities a and a, then also k tan (a + WTT) = -, ftt n denoting a positive integer. The second of equations (1) is satisfied, therefore, if we assume COS)/' = a + mr, and make r tend towards zero by increasing n indefinitely. If we substitute L- for r conformably with equation (2), we get for which, since cos \l> differs from 1 only by an infinitesimal of the second order when \j/ and r are infinitesimals of the first order, we can also write (3) t = * a + mr INF.
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