The power series 00 D(z) = Σ %3D n=1 converges absolutely on |z| < 1 with the radius of convergence 1 and satisfies the relations D(1) = 5(2) and D(-1) = -{(2)/2. We have ² log (1-z) dz, D(z) = - from which D(z) has an analytic continuation to the whole complex plane except for the half line [1, 00) by taking the principal branch for the logarithm log (1 - z). Show then the functional equation 1 D(-2) = 2D(-1) – → log? z + and consider the limit as z →-1 in the upper half plane.
The power series 00 D(z) = Σ %3D n=1 converges absolutely on |z| < 1 with the radius of convergence 1 and satisfies the relations D(1) = 5(2) and D(-1) = -{(2)/2. We have ² log (1-z) dz, D(z) = - from which D(z) has an analytic continuation to the whole complex plane except for the half line [1, 00) by taking the principal branch for the logarithm log (1 - z). Show then the functional equation 1 D(-2) = 2D(-1) – → log? z + and consider the limit as z →-1 in the upper half plane.
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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Transcribed Image Text:The power series
00
D(z) =
Σ
n²
n=1
converges absolutely on |z| < 1 with the radius of convergence 1 and satisfies
the relations D(1) = 3(2) and D(-1) = -3(2)/2. We have
log (1- z)
dz,
D(z) =
from which D(z) has an analytic continuation to the whole complex plane
except for the half line [1, c∞0) by taking the principal branch for the logarithm
log (1
- z). Show then the functional equation
1
D
+ D(-z) =D 2D(-1) - 등 log? z
2
and consider the limit as z → -1 in the upper half plane.
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