I was solving some integration questions, then one question arised in my mind that "Do the integration of complex no. possible?"
If yes, then what is $\int i dx$?
Definite integration is the area under curve of the graph but the above graph cannot be plotted on real plane.
I want someone to please clear my doubts regarding integration. Thanks!!!
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sgrmshrsm7
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$i$ is a constant, so it can be moved outside of the integral. – Qudit Mar 17 '17 at 19:44
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1answer is $=ix+C$ – haqnatural Mar 17 '17 at 19:44
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And what about definite integration. – sgrmshrsm7 Mar 17 '17 at 19:45
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@haqnatural Is $C \in \mathbb{C}$ or is $C\in\mathbb{R}$? – mrnovice Mar 17 '17 at 19:46
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1In general for continuous functions $\int_{a}^b u(x)+iv(x)=\int_{a}^b u(x)+i\int_{a}^b v(x)$.Also haqnatural but $(ix+5i)'=i$. – kingW3 Mar 17 '17 at 19:53
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Possible duplicate of How do you integrate imaginary numbers? – scrappedcola Mar 17 '17 at 19:54
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https://terrytao.wordpress.com/2016/09/27/246a-notes-2-complex-integration/ – Mar 17 '17 at 19:56
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If the function has a complex derivative and is defined on the entire complex plane, then it has an antiderivative that's also defined on the entire complex plane. All polynomials have antiderivatives, for example. However, $\frac1x$ (which has a complex derivative, $-\frac1{x^2}$, but isn't defined at $x=0$) does not have an antiderivative defined everywhere (ignoring zero). It does have what are called "local antiderivatives," though. – Akiva Weinberger Mar 17 '17 at 20:28
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Did you see an answer you would accept? – kleineg Mar 20 '17 at 03:25
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The simple answer is that $i$ is a constant, so $\int i \,dx = i x + C$
The more complete answer is that "area under curve of the graph" doesn't really make sense for what you are doing when you say $\int i \,dx$, you can integrate a complex variable $z=x +iy$ over a contour. I would check out contour integration here, it might explain things a little better.
kleineg
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