Principle values of log

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

Write the expression $\log\left(\simplify{{a1}*i^(1/{b1})}\right)$ in the form $u+iv$ , where $u$ and $v$ are to be determined, and $v$ depends on some index $n=0,\pm 1,\pm 2,\ldots$ , say.

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			Name	Type	Generated Value

a1	integer	1
u	number	0.000
b1	integer	5
tol	number	0.001

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Generated value: `integer`

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1. Gap Gap 0. Number entry
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2. Gap Gap 1. Mathematical expression
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$u=$ Gap 0.. (Enter your answer to 3d.p.)

$v=$ Gap 1.. (Do not use decimals in your answer. Use $n$ for the index, and if you need to use $\pi$ in your answer, write pi.)

Advice

First write the expression $z=\log\left(\simplify{{a1}*i^(1/{b1})}\right)$ as

$z=\ln(\var{a1})+\log\left(\simplify{i^(1/{b1})}\right),$

then

$z=\ln(\var{a1})+\frac{1}{\var{b1}}\log(i).$

Now

$\log(i)=\operatorname{Log}(i)+2n\pi i, \quad n=0,\pm 1,\pm 2,\ldots,$

where $\operatorname{Log}(i)$ is the principal value of $\log(i)$ .

Then

$\operatorname{Log}(i)=\ln\lvert i\rvert+i\operatorname{Arg}(i)=0+\frac{\pi}{2}i,$

$\log(i)=0+\frac{\pi}{2}i+2n\pi i=\frac{(1+4n)\pi i}{2}, \quad n=0,\pm 1,\pm 2,\ldots.$

Therefore

$z=\log\left(\simplify{{a1}*i^(1/{b1})}\right)=\ln(\var{a1})+\frac{(1+4n)\pi i}{\var{2*b1}}, \quad n=0,\pm 1,\pm 2,\ldots.$

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This question is used in the following exam:

Complex variables by Newcastle University Mathematics and Statistics in Content created by Newcastle University.

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

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Principle values of log

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`