Limits involving logarithm and argument in the complex plane Log((2/n)+2i) as n\to\infty

Cale Terrell

Cale Terrell

Answered question

2022-10-22

Limits involving logarithm and argument in the complex plane

 

Log((2/n)+2i) as n

Log(2+(2i/n)) as n

Arg((1+i)/n) as n

(Arg(1+i))/(n) as n

 

For the Log questions, I am getting (iπ)/2+log(2) for the first problem, then for the second I am getting only log(2). Because the Log's in the questions are capitalized, I think I may have to add on 2πik to each of the answers. Is that correct?

for the last two problems (the Arg problems) I got zero for both because as n goes to infinity the n is the denominator for each so I thought they probably each go to zero. but also, the Arg is capitalized here as well, so I am getting the feeling I am doing these wrong. Can anybody help? Thanks!

Answer & Explanation

Kaylee Evans

Kaylee Evans

Beginner2022-10-23Added 20 answers

The meaning of capitalized names such as Log varies by source. I assume that Log has been defined so that it's continuous at 2i and at 2; this is the case for the common definitions I'm familiar with. Check your definition. Then
Log((2/n)+2i)Log(2i) as n
Log(2+2i/n)Log(2) as n
Arg((1+i)/n)=Arg(1+i) for all n; this is a constant sequence. Argument of a complex number is not affected by scaling.
(Arg(1+i))/n0 since numerator does not depend on n, while the denominator grows indefinitely.

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