Calculating Residue at -2: Math Methods w/ Arfken et al.

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Homework Statement
Calculate the residue of the complex function ## f(z) = \dfrac{\ln z}{z^2 + 4} ## at ## z = 2 e^{i \pi} ##
Relevant Equations
## a = \lim_{z \rightarrow z_0} \left[ \left(z_0 - z\right) f(z) \right] ##
This question is given as an example in the book by
Arfken, Weber, Harris, Mathematical Methods - a Comprehensive Guide, Seventh Edition.

It is solved as below attached in the image.

Can someone point it out how they proceed with calculations ? I do not seem to get their calculation.
I am aware ## \ln z ## is a multivalued function. But at this point I do not know things about Branch points and etc.

According to my understanding the function is not singular at point ## z=2 e^{i \pi} =−2 ## . So why they have used limits ?

Am I missing something ? Please help.

Thanks.
 

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Even then, in numerator, it should be ## \ln 2 + i \dfrac{\pi}{2} ## right ?
 
fresh_42 said:
I guess there is a typo and it should be ##z_1=2e^{i \pi /2}.##
Or the denominator was supposed to be ##z^2-4##. In either case, the example is all messed up.
 
curious_mind said:
Even then, in numerator, it should be ## \ln 2 + i \dfrac{\pi}{2} ## right ?
Yes, ...
vela said:
Or the denominator was supposed to be ##z^2-4##. In either case, the example is all messed up.
... and yes. At least, ##2e^{i \pi /2}## is a singularity. However, ##z=-2## would be faster to solve.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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