Complex argument (continued fraction)

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In analysis, the complex argument θ = arg(z) is commonly defined as an angle, often in terms of the inverse tangent function, or the inverse cosine. In a purely formal and perfectly rigorous treatment of the complex numbers, such reliance on geometric intuition should be avoided.[1] Here is one way the complex argument θ, −π < θ ≤ π, can be defined without reference to any geometrical or trigonometric construction.

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[edit] Formal definition of arg(z)

By convention, arg(0) is undefined, since there is no complex number z for which ez = 0.[2] We define arg(z) = π for real z < 0. When z is not a real number ≤ 0, we define

z = x + iy \qquad |z| = \sqrt{x^2 + y^2} \qquad \theta = \arg(z) = -i\log\frac{z}{|z|} \,

where −π < θ < π.[3] This is the principal value of the argument function arg(z).

It can be shown that the infinite continued fraction

\log\frac{1+w}{1-w} = \cfrac{2w}{1 - \cfrac{w^2}{3 - \cfrac{4w^2}{5 - \cfrac{9w^2}{7 - \cfrac{16w^2}{\ddots}}}}}\,

converges uniformly when w is not a real number, and also when w is real and |w| < 1.[4] The logarithm function thus defined is holomorphic on the domain of uniform convergence.

Setting

\frac{z}{|z|} = \frac{1+w}{1-w} \quad \Rightarrow \quad w = \frac{z-|z|}{z+|z|}

now completes the formal definition of arg(z) without reference to any geometrical construction.

[edit] Historical context

Euler's formula connecting this continued fraction with an infinite logarithmic series was first published in 1748. Euler was not particularly concerned with procedural niceties, and mathematicians today generally regard his work as original, and very useful, but not entirely rigorous.[5]

Gauss found another way to derive this continued fraction from the hypergeometric series in 1813. Gauss proved a general formula, but he did not have the analytical tools necessary to verify the uniform convergence of this particular continued fraction.[6]

Up through the time of Gauss, mathematicians generally did not draw a sharp distinction between geometrical and analytical concepts, and used these frameworks almost interchangeably. This practice continues in many modern presentations of mathematical ideas – charts and graphs are freely used to illustrate the concepts of analysis.

In the mid-19th century mathematicians like Cauchy and Weierstrass began insisting on more rigorous proofs. This emphasis on rigor eventually led to Hilbert's program of formalism in mathematics, and that stimulated a desire to state the concepts of mathematical analysis and the concepts of geometry without mixing them together.

[edit] See also

[edit] Notes

  1. ^ (Whittaker and Watson, 1927, Appendix). W. and W. provide a purely analytical approach to this problem, but instead of dealing directly with the natural logarithm, their analysis begins with Euler's formula for the exponential function.
  2. ^ See Picard theorem.
  3. ^ Equality between the symbol π as used here and the π used in Euclidean geometry can also be demonstrated analytically.
  4. ^ (Wall, 1948, p. 343) attributes this particular continued fraction to Euler.
  5. ^ (Euler, 1748; cited in Wall, 1948).
  6. ^ (Gauss, 1813; cited in Wall, 1948).

[edit] References

  • Leonhard Euler, Introductio in analysin infinitorum, Vol. I, Chapter 18 (1748).
  • Karl Friedrich Gauss, Werke, Vol. 3, pp. 134-138 (1813).
  • H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948; reprinted (1973) by Chelsea Publishing Company ISBN 0-8284-0207-8.
  • E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927.