Let a be an arbitrary positive irrational number. Show that the unit circle |z| = 1 is the natural boundary of the power series 00 f(z) = [[an]z". n=1 This is due to Hecke (1921). The function f(z) is sometimes called the Hecke- Mahler function, being related to Mahler's function of two variables: co [an] Fa(w,z) = Σ Σ w"z" n=1 m=1 by the relation F(1,z) = f(z). Various arithmetical properties of the values of f(z) can be studied by virtue of certain functional equations satisfied by Fa(w, z), F1/a (z, w) and Fk+a (w, z). The author (1982) encountered this function in the study of a mathematical neuron model as a special case of Caianiello's equation (1961).
Let a be an arbitrary positive irrational number. Show that the unit circle |z| = 1 is the natural boundary of the power series 00 f(z) = [[an]z". n=1 This is due to Hecke (1921). The function f(z) is sometimes called the Hecke- Mahler function, being related to Mahler's function of two variables: co [an] Fa(w,z) = Σ Σ w"z" n=1 m=1 by the relation F(1,z) = f(z). Various arithmetical properties of the values of f(z) can be studied by virtue of certain functional equations satisfied by Fa(w, z), F1/a (z, w) and Fk+a (w, z). The author (1982) encountered this function in the study of a mathematical neuron model as a special case of Caianiello's equation (1961).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let a be an arbitrary positive irrational number. Show that the unit circle
|z| = 1 is the natural boundary of the power series
f(z) =
[an]z".
n=1
This is due to Hecke (1921). The function f(z) is sometimes called the Hecke-
Mahler function, being related to Mahler's function of two variables:
00 [an]
Fa(w,z) = Σ Σ wme"
n=1 m=1
by the relation F(1, z) = f(z). Various arithmetical properties of the values
of f(z) can be studied by virtue of certain functional equations satisfied by
Fa (w, z), F1/a (z, w) and Fk+a (w, z).
The author (1982) encountered this function in the study of a mathematical
neuron model as a special case of Caianiello's equation (1961).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde14fa0d-eeba-47e2-9edf-6e36315a5d4e%2Fcb4b2ee5-74f2-403e-97c4-d46daab6fb92%2Fxeuti3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a be an arbitrary positive irrational number. Show that the unit circle
|z| = 1 is the natural boundary of the power series
f(z) =
[an]z".
n=1
This is due to Hecke (1921). The function f(z) is sometimes called the Hecke-
Mahler function, being related to Mahler's function of two variables:
00 [an]
Fa(w,z) = Σ Σ wme"
n=1 m=1
by the relation F(1, z) = f(z). Various arithmetical properties of the values
of f(z) can be studied by virtue of certain functional equations satisfied by
Fa (w, z), F1/a (z, w) and Fk+a (w, z).
The author (1982) encountered this function in the study of a mathematical
neuron model as a special case of Caianiello's equation (1961).
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