Section 6

Convergence in Separated Regions

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Problem 3.259

The series

z1+z+z2(1+z)(1+z2)+z4(1+z)(1+z2)(1+z4) \frac{z}{1+z} + \frac{z^{2}}{(1+z)(1+z^{2})} + \frac{z^{4}}{(1+z)(1+z^{2})(1+z^{4})} +z8(1+z)(1+z2)(1+z4)(1+z8)+⋯ + \frac{z^{8}}{(1+z)(1+z^{2})(1+z^{4})(1+z^{8})} + \cdots

converges uniformly in any domain that lies either entirely inside or entirely outside the unit circle and its sum is zz or 11 according as ∣z∣<1\left|z\right|<1 or ∣z∣>1\left|z\right|>1 [problem 14 of Part I].

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Problem 3.260

Let α\alpha denote an arbitrary constant, α≠0\alpha \neq 0. The series

1+∑n=1∞α(α+n)n−1xne−nxn! 1 + \sum _{n=1}^{\infty } \frac{\alpha (\alpha +n)^{n-1}x^{n}e^{-nx}}{n!}

converges uniformly for all positive values of xx; it represents the function eαxe^{\alpha x} for 0≤x≤10\leq x\leq 1 and a different analytic function for 1<x<∞1<x<\infty.

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Problem 3.261

The sequence of functions f1(z),f2(z),…,fn(z),…f_{1}(z),f_{2}(z),\ldots ,f_{n}(z),\ldots

fn(z)=[n1]1z+[n2]2z+⋯+[nn]nzn(1z−1+2z−1+⋯+nz−1),n=1,2,3,…, f_{n}(z) = \frac{\left[\dfrac {n}{1}\right]1^{z} + \left[\dfrac {n}{2}\right]2^{z} + \cdots + \left[\dfrac {n}{n}\right]n^{z}}{n(1^{z-1}+2^{z-1}+\cdots +n^{z-1})}, \quad n=1,2,3,\ldots ,

converges uniformly in any finite domain that does not contain the imaginary axis.

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Problem 3.262

Let

α>0,β>0,α+β=1, \alpha >0, \quad \beta >0, \quad \alpha +\beta =1,

and put

φ(z)=αz+β1z. \varphi (z) = \alpha z + \beta \frac{1}{z}.

The sequence of iterated functions

φ(z),φ[φ(z)],φ{φ[φ(z)]},… \varphi (z), \quad \varphi [\varphi (z)], \quad \varphi \{ \varphi [\varphi (z)]\} , \quad \ldots

converges to +1+1, when Re(z)>0\mathrm{Re}\left(z\right)>0, to −1-1 when Re(z)<0\mathrm{Re}\left(z\right)<0 and diverges when Re(z)=0\mathrm{Re}\left(z\right)=0.

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