How do I perform this complex integration?

Question

Evaluate the complex integration $(z^2 + 3z)$ wrt $z$ along the circle $|z| = 2$, from $(2,0)$ to $(0,2)$ in a counterclockwise direction.

As far as I understand, this can be solved by taking $x = 2 \cos \theta$, $y = 2 \sin \theta$, and then integrating wrt $\theta$ from $0$ to $π/2$.

But on making these substitutions, the integration becomes quite lengthy and clumsy. Is there any other way to solve this, which I might be missing right now?

Try a contour integral, using (0,0) to (2,0) and (0,2) to (0,0) as additional sides. and note that the contour integral has no poles. — Andrei, Nov 17 '18 at 03:33

Martin Argerami · Answer 1 · 2018-11-17T06:08:26.257

2

On your circle, $z=2e^{it}$, $0\leq t\leq\pi/2$. Then your integral is (never forget to include the change of variable, $dz=2ie^{it}\,dt$) \begin{align} \int_0^{\pi/2} ((2e^{it})^2+6e^{it})\,2ie^{it}\,dt &=2i\int_0^{\pi/2}(4e^{3it}+6e^{2it})\,dt =2i\left[ \left.\frac{4e^{3it}}{3i}+\frac{3e^{2it}}{i}\right|_0^{\pi/2}\right]\\ \ \\ &=-\frac {8i}3-6-\frac83-6=-\frac {44}3-\frac {8i}3. \end{align}

edited Nov 17 '18 at 06:08

answered Nov 17 '18 at 03:33

Martin Argerami

217,281

2

Yes. Now I get it. Instead of writing z as x + iy, I should write it in the r e^it form, which makes the task simpler. – Jasmine Nov 17 '18 at 03:46
@MartinArgerami You lost a factor of $e^{it}$ going between the two integral expressions. – epimorphic Nov 17 '18 at 05:04
Indeed, thanks for noticing! – Martin Argerami Nov 17 '18 at 05:41
+1, but still some errors. The final answer should be $$-\frac{44}{3}-\frac{8}{3}i$$ – quasi Nov 17 '18 at 05:47
Indeed. Thanks. – Martin Argerami Nov 17 '18 at 06:08
Yes. This is the final answer given in my book. Thank you – Jasmine Nov 18 '18 at 18:44

Mohammad Riazi-Kermani · Answer 2 · 2018-11-17T05:56:13.657

2

$$ \int _{2}^{2i} z^2+3z dz =(1/3)z^3 +(3/2) z^2 |_2^{2i} = (-44/3)-(8/3)i$$

edited Nov 17 '18 at 05:56

answered Nov 17 '18 at 04:01

Mohammad Riazi-Kermani

69,891

This is easy ! I had thought of this earlier, but what confused me was the circle given in question. – Jasmine Nov 17 '18 at 04:18
@Jasmine: This is because of "The Fundamental Theorem of Calculus". – Yadati Kiran Nov 17 '18 at 04:24
@epimorphic: I do agree that the Fundamental Theorem of Calculus cannot be applied to complex contour integrals but its the most efficient way to get the result provided the domain is simply connected and the function is analytic on the domain – Yadati Kiran Nov 17 '18 at 05:37
@Mohammad Riazi-Kermani: Your integrand should be $z^2+3z$. – quasi Nov 17 '18 at 05:49
@YadatiKiran Oops, you're right, I really need to review my complex analysis... *slinks back into corner* – epimorphic Nov 17 '18 at 05:50
@quasi Thanks for the comment. I fixed my error. – Mohammad Riazi-Kermani Nov 17 '18 at 05:57
@epimorphic: A kind suggestion: You may leave your comments undeleted so that it may help people with similar doubts to correct their arguments. – Yadati Kiran Nov 17 '18 at 06:01

How do I perform this complex integration?

2 Answers2