@@MathSolvingChannelCould you please tell me what the value of Phi of C2 and C4 paths are in number? And how do you get Phi of C2 and C4 path from? Or choose Phi between 0 to 2Pi?
Noice!! Another way is sub z=e^x so you get integral e^(ax)/(1+e^x) from -inf to inf and solve with contour integration over the rectangle -inf to inf (real axis) and 0 to 2pi(Im axis).😀
Thank you for sharing this :) This is the technique called "conformal mapping". Intuitively, exp-function and log-function they are mapping the region switching between rectangle and circle (or sector) 😉
Hi Just like in C3 we took ze^(2pi*i), why dont we do the same (i.e. z -> ze^(pi*i) = -z)when calculating similar integral of sinx/x (in case of tunnel contour)
Yes, if the integral limit is from 0 to 1, then you can do the series expansion 1/(1+x)=sum (-1)^n*x^n, so the integral became sum (-1)^n* integral x^(a+n-1) dx from 0 to 1, this integral is simple, after integrate it you get sum (-1)^n/(n+a) where n is from 0 to infinity as the result 😉
@@MathSolvingChannel thank you, i have another question: do you think you can find a closed form for sum[1/(a*n+1)^2] where a is just a real number but a =/= 0
@@xulq It depends on what do you mean by "closed form". You can use digamma function to express it, only to "express" it. If you want a closed form such as some combinations of pi, e, etc, then answer is No.
@@xulq You are welcome :) This is like the Gaussian integral (just an example for analogy), if the upper limit is not infinity, but some finite number, such as integral e^(-x^2) from 0 to 1, if we accept this expression as a "closed form", then it is just an expression. But we can't express it into the combination of those famous math constant.
Great.
I love complex analysis and ... Of course, I love this video too.
Thank you so much.
Glad you liked it!
@@MathSolvingChannelCould you please tell me what the value of Phi of C2 and C4 paths are in number? And how do you get Phi of C2 and C4 path from? Or choose Phi between 0 to 2Pi?
Really good description of the solution. Thanks!
this integral is tough, great video!
Glad you think so!
Great Video!
Glad you enjoyed it!
great video! this integral is very useful!
Glad it was helpful!
Great video, thanks :)
Noice!! Another way is sub z=e^x so you get integral e^(ax)/(1+e^x) from -inf to inf and solve with contour integration over the rectangle -inf to inf (real axis) and 0 to 2pi(Im axis).😀
Thank you for sharing this :) This is the technique called "conformal mapping". Intuitively, exp-function and log-function they are mapping the region switching between rectangle and circle (or sector) 😉
@@MathSolvingChannel good to know,i saw this in complex analysis book of princeton lectures.(great book btw)
@@yoav613 Great! 😆
Excellent
Looks very advanced math.
Why keyhole is needed here? Test case?
What is the result if its not complex analysis case?
Result is independent of what method you choose, as long as you did it correctly.
Why don't the integrals along paths C1 and C3 sum to 0? Intuitively, it seems that they should, although I understand your explanation.
Hi
Just like in C3 we took ze^(2pi*i), why dont we do the same (i.e. z -> ze^(pi*i) = -z)when calculating similar integral of sinx/x (in case of tunnel contour)
Why not plug in e^(2*pi*i)=1 at time=2.34 without carrying it further?
can you do this for the same integrand but integrated from 0 to 1 instead?
Yes, if the integral limit is from 0 to 1, then you can do the series expansion 1/(1+x)=sum (-1)^n*x^n, so the integral became sum (-1)^n* integral x^(a+n-1) dx from 0 to 1, this integral is simple, after integrate it you get sum (-1)^n/(n+a) where n is from 0 to infinity as the result 😉
@@MathSolvingChannel thank you, i have another question:
do you think you can find a closed form for sum[1/(a*n+1)^2] where a is just a real number but a =/= 0
@@xulq It depends on what do you mean by "closed form". You can use digamma function to express it, only to "express" it. If you want a closed form such as some combinations of pi, e, etc, then answer is No.
@@MathSolvingChannel aww okay thank you :v
@@xulq You are welcome :) This is like the Gaussian integral (just an example for analogy), if the upper limit is not infinity, but some finite number, such as integral e^(-x^2) from 0 to 1, if we accept this expression as a "closed form", then it is just an expression. But we can't express it into the combination of those famous math constant.
I think you should relefect how many videos you upload
what?😵
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😉
@@MathSolvingChannel u know who she is
@@MathSolvingChannel math Malta 🇲🇹,,I have emailed that trigo questions
@@anonymousgawd..3047 who?
@@anonymousgawd..3047 no, I didn't receive it yet