Today we will try to resolve the indefinite integral...
(1)
... using the residue theorem. This theorem gives the integral of a complex variable function f(*) along a closed path C as…
(2)
… where is the residue of f(*) in one singularity of order n internal to C…
(3)
The f(*) we choose is...
(4)
... and the path C is indicated in figure as ABCDEF...
Since there are no singulatities internal to the path is...
(5)
First we will consider the integral along ABC, to which we can apply the so called 'Jordan's Lemma' that says that if there are two constants M and k>1 so that is for , then is...
(6)
... and that for the f(*) given in (4) is verified for and ... very well!... In that case, taking into account (5) and (6), we can write...
(7)
... or in similar way...
(8)
... and this limit will be computed in next post...
Kind regards
In my previous post the attempt to compute the definite integral…
(1)
... has failed because the function I proposed didn’t satisfy the ‘Jordan’s Lemma’. So I will correct the post and please the moderators, if possible, to delete my previous post in order to avoid misunderstanding…
The so called ‘Residue Theorem’ gives the integral of a complex variable function f(*) along a closed path C as…
(2)
… where is the residue of f(*) in one singularity of order n internal to C…
(3)
The f(*) we choose is the function proposed by Fantastic...
(4)
... and the path C is indicated in figure as ABCDEF...
Since there are no singulatities internal to the path is...
(5)
First we will consider the integral along ABC, to which we can apply the so called 'Jordan's Lemma' that says that if there are two constants M and k>1 so that is for , then is...
(6)
... and that for the f(*) given in (4) is verified for and ... very well!... In that case, taking into account (5) and (6), we can write...
(7)
... or in similar way...
(8)
... and this limit will be computed in next post...
Kind regards
Now we perform the final step, that is the computation of...
(1)
... where...
(2)
... and the path AB is represented in the figure...
For the computation of (1) we set , so that is and the integral becomes...
(3)
Using the Taylor expansion of exponential...
(4)
... we obtain the function g(*) that must be integrated in the form...
(5)
... and what really matters is that...
(6)
... so that...
(7)
Kind regards