ne complex logar ubo pur 80T JO and the complex square root. What is the domain U C C of these two functions. ate the ) Starting from the definitions, establish the identity Log(Sqrt(2)) = ;Log(z), z e U. %3D

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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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(a) State the definition of Log and Sqrt, the principal branches of the complex logarithm
and the complex square root. What is the domain U C C of these two functions.
(b) Starting from the definitions, establish the identity Log(Sqrt(z)) = } Log(2), z E U.
(c) Let U1 = C\ [0, 00). Describe the branch Log, of the logarithm valid in U1
(d) Explain why the identity Log,(Sqrt(z)) = 1/2 Log,(2) does not hold in general. Find
a maximal domain for which this identity is valid.
Transcribed Image Text:(a) State the definition of Log and Sqrt, the principal branches of the complex logarithm and the complex square root. What is the domain U C C of these two functions. (b) Starting from the definitions, establish the identity Log(Sqrt(z)) = } Log(2), z E U. (c) Let U1 = C\ [0, 00). Describe the branch Log, of the logarithm valid in U1 (d) Explain why the identity Log,(Sqrt(z)) = 1/2 Log,(2) does not hold in general. Find a maximal domain for which this identity is valid.
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