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0 as n -.. "" and 1f3 n I belongs, for each n, to the interval (R 1, R2) and a sequence lfn(z)l of functions fn(z) = z + '" that are regular in 1zl < 1 such that, for every z in Iz\ < 1, we have f n (z) f- a n for every n and ep n (z) f n(z) = f3 n for no more than p - 1 values of z.

And 1f3 n I belongs, for each n, to the interval (R 1, R2) and a sequence lfn(z)l of functions fn(z) = z + '" that are regular in 1zl < 1 such that, for every z in Iz\ < 1, we have f n (z) f- a n for every n and ep n (z) f n(z) = f3 n for no more than p - 1 values of z. But the functions = [f'(z) - a n ]/(f3 n - a n ) do not assume the value 0 in n Izi < 1 and they 't" ~ :lit *:~ assume the value 1 in that disk at no more than p - 1 points. Consequently, 1) Landau [1922]. 1) Mantel 2) A more profound study of me modulat function led Bermant [1944] to a number of [1933].

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Algebraic topology: homology and cohomology by Andrew H. Wallace


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