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ln ( 10.75 + 6.98i )

difficulty advanced
Guest Jan 15, 2015

Best Answer 

 #1
avatar+18715 
+10

ln ( 10.75 + 6.98i )

$$\small{\text{
$z=10.75+6.98\ i \quad \ln{(z)} = ?$
}}\\
\small{
Solution:
\boxed{ln{(z)} = \ln{(|z|)} + i*(arg(z)+2\pi k ) }
}
\\\\
\small{\text{
I. $ |z| = \sqrt{10.75^2+6.98^2} = 12.8172891050 $
}}\\
\small{\text{
II. $ \phi\ensurement{^{\circ}} = \tan^{-1}{(\frac{6.98}{10.75} )} = 32.9957575073\ensurement{^{\circ}} $
}}\\
\small{\text{
III. $ arg(z) = \phi \ensurement{^{\circ}} *
\frac{ \pi }{ 180\ensurement{^{\circ}} }+2\pi k = 0.57588460769 + 2\pi k$
}}\\\\
\small{\text{
$
\ln{(z)} = \ln{(12.8172891050 )}+i*(0.57588460769+2\pi k)
$
}}\\
\small{\text{
$
\boxed{ \ln{(10.75+6.08\ i)} = 2.55079497086+(0.57588460769+2\pi k) \ i } \quad k=0,1,2\dots
$
}}$$

heureka  Jan 15, 2015
Sort: 

3+0 Answers

 #1
avatar+18715 
+10
Best Answer

ln ( 10.75 + 6.98i )

$$\small{\text{
$z=10.75+6.98\ i \quad \ln{(z)} = ?$
}}\\
\small{
Solution:
\boxed{ln{(z)} = \ln{(|z|)} + i*(arg(z)+2\pi k ) }
}
\\\\
\small{\text{
I. $ |z| = \sqrt{10.75^2+6.98^2} = 12.8172891050 $
}}\\
\small{\text{
II. $ \phi\ensurement{^{\circ}} = \tan^{-1}{(\frac{6.98}{10.75} )} = 32.9957575073\ensurement{^{\circ}} $
}}\\
\small{\text{
III. $ arg(z) = \phi \ensurement{^{\circ}} *
\frac{ \pi }{ 180\ensurement{^{\circ}} }+2\pi k = 0.57588460769 + 2\pi k$
}}\\\\
\small{\text{
$
\ln{(z)} = \ln{(12.8172891050 )}+i*(0.57588460769+2\pi k)
$
}}\\
\small{\text{
$
\boxed{ \ln{(10.75+6.08\ i)} = 2.55079497086+(0.57588460769+2\pi k) \ i } \quad k=0,1,2\dots
$
}}$$

heureka  Jan 15, 2015
 #2
avatar+91051 
0

Thanks Heureka, that looks impressive  

Melody  Jan 15, 2015
 #3
avatar+26329 
+5

Here's an alternative approach (I've just considered the principal solution):

 

log of complex number

.

Alan  Jan 16, 2015

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