8. We define a pole zo of f(z) to be a critical point of f(z) of order kif zo is a critical point of 1/f(z) of order k. We define z = ∞ to bea critical point of f(2) of order k if w = 0 is a critical point ofg(w) = f(1/w) of order k. Show that with this definition, a pointVIII The Logarithmic Integralzo € C* is a critical point of order k for a meromorphic function f(2)if and only if there are open sets U containing zo and V containingwo = f(z0) such that each w e V, w # wo, has exactly k + 1preimages in U. Remark. We say that f(z) is a (k + 1)-sheetedcovering of f-1(V\{wo}) nU over V\{wo}.

We define a pole zo of f(2) to be a critical point of f(z) of order k if zo is a critical point of 1/f(z) of order k. We define z = oo to be a critical point of f(z) of order k if w = 0 is a critical point of g(w) = f(1/w) of order k. Show that with this definition, a point VIII The Logarithmic Integral z0 € C* is a critical point of order k for a meromorphic function f(2) if and only if there are open sets U containing zo and V containing wo = f(zo) such that each w e V, w # wo, has exactly k + 1 preimages in U. Remark. We say that f(2) is a (k + 1)-sheeted covering of f-'(V\{wo})nU over V\{wo}.

We define a pole zo of f(2) to be a critical point of f(z) of order k if zo is a critical point of 1/f(z) of order k. We define z = oo to be a critical point of f(z) of order k if w = 0 is a critical point of g(w) = f(1/w) of order k. Show that with this definition, a point VIII The Logarithmic Integral z0 € C* is a critical point of order k for a meromorphic function f(2) if and only if there are open sets U containing zo and V containing wo = f(zo) such that each w e V, w # wo, has exactly k + 1 preimages in U. Remark. We say that f(2) is a (k + 1)-sheeted covering of f-'(V\{wo})nU over V\{wo}.

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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Question

8. We define a pole zo of f(z) to be a critical point of f(z) of order k
if zo is a critical point of 1/f(z) of order k. We define z = ∞ to be
a critical point of f(2) of order k if w = 0 is a critical point of
g(w) = f(1/w) of order k. Show that with this definition, a point
VIII The Logarithmic Integral
zo € C* is a critical point of order k for a meromorphic function f(2)
if and only if there are open sets U containing zo and V containing
wo = f(z0) such that each w e V, w # wo, has exactly k + 1
preimages in U. Remark. We say that f(z) is a (k + 1)-sheeted
covering of f-1(V\{wo}) nU over V\{wo}.

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ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated