Category Archives: Complex Variables

On the Euclidean Growth of Entire Functions

In Mapping properties of analytic functions on the unit disk, Proceedings of the American Mathematical Society, Vol. 135, N. 9 (2007), 2893-2898. I show that there is a universal constant such that whenever is analytic in the unit disk and … Continue reading

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Square-roots of Complex Numbers

If is a complex number in polar coordinates, with say , then one of its square-roots is . But what if one wants to avoid using the exponential function? This trick was related to me by my colleague Bob Burckel. … Continue reading

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Complex Numbers

A complex number can be written in Cartesian coordinates as , where is the real part and is the imaginary part. Or it can be written in polar coordinates as , where is the absolute value and is the point … Continue reading

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A little-known definition of Hardy spaces

In the paper by Sedleckiń≠, A. M. An equivalent definition of the spaces in the half-plane, and some applications. (Russian) Mat. Sb. (N.S.) 96(138) (1975), 75–82, 167, the following is proved: given an analytic function on the upper-half plane consider … Continue reading

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Global harmonic miracle

That harmonic functions would satisfy the mean value property for infinitesimally small circles is intuitively somewhat clear. But that this property should persist on circles of arbitrary radius is quite miraculous. To recall, if is harmonic on a domain and … Continue reading

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Qual-type problem

Problem: Let be entire and suppose that for all : Show that . Solution: For let On one hand, using the fact that we get that as , for each . And on the other hand, direct integration shows that … Continue reading

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