Evaluate $\int_{\mathcal{C}}\frac{\sin z}{z-\pi/2}\,\mathrm{d}z$

Question

Problem.¹⁾ Evaluate $\displaystyle \int_{\mathcal{C}}\frac{\sin z}{z-\pi/2}\,\mathrm{d}z$, given that $\mathcal{C} : |z| = \pi/2$

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score 2 · Accepted Answer · answered May 17 '22 at 08:19

2

Use Cauchy's integral formula.

$$\int_{|z|=a} \frac{f(z)}{z-a}dz = 2\pi i f(a) $$

For your question $f(z) = sin(z)$

So answer to your question is $2\pi i sin(\frac{\pi}{2}) = 2\pi i$

answered May 17 '22 at 08:19

Fire

2

But the contour $C$ contains a discontinuous point $\frac{\pi}{2}$. – Muses_China May 17 '22 at 09:13
The Cauchy's integral formula https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula requires $a$ to be inside of the contour, not on the contour. Here the integrand is unbounded, so it is not integrable. – Muses_China May 18 '22 at 06:53
Yes, didn't noticed that. @J.D the answer is probably wrong. – Fire May 18 '22 at 07:32

1 Answers1