Section 31- 32, The Logarithmic function Flashcards

1
Q

let $0 \not = z \in C$

If $e^w = z$ then

A

$w = ln|z| + i arg(z) = ln|z| + i(Arg(z) + 2 \pi i) , n \in Z$

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2
Q

Multiple valued logarithmic function log$z$ is defined as

A

log $z = ln |z| + i arg(z) = ln|z| + i(Arg(z) + 2 \pi i) , n \in Z$

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3
Q

Clearly $e^{log(z)}=$

A

z

for $z \not = 0, z \in C)

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4
Q

log($e^z$) = ?

A

$z + 2 n \pi i, n \in Z$

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5
Q

Log is a single valued function onwhat domain

How is Log z into log z

A

domain of Log z is $\mathbb{C}${0}

and

$log z = Log z + 2 n \pi i, n\in Z$

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6
Q

When and how does the complex logarithm turn into the real case

A

If $z = x + i 0, x>0$ then $|z| = x$ and Arg z = 0

so that Log z = ln x reduces to teh real logarithmic function

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7
Q

log$(z_1 z_2) = $

A

$log z_1 + log z_2$

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8
Q

log $\frac{z_1}{z_2} = $

A

$log z_1 - log z_2$

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9
Q

$z^n =$

A

$e^{nlogz} \qquad n \in Z$

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10
Q

$z^{1/n}=$

A

$e^{(1/n)logz} \qquad n\in Z$

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11
Q

log$(z_1 z_2) = $

A

NOT
$Log z_1 + Log z_2$

UNLESS
Re$(z_1) > 0$ and Re$(z_2)>0$

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