a)0b)π/2c)1/5d)π/3Correct answer is option 'B'. Can you explain this answer?

Question

a)0b)&pi;/2c)1/5d)&pi;/3Correct answer is option 'B'. Can you explain this answer?

Answer

Explain the questions

Answer

Concept:

Cauchy's Integral Formula

If f(z) is analytic within and on a closed curve and if a is any point within C, then:

∫C⁡f(z)z−adz=2πif(a)

Analysis:

Given:

f(z)=∫1z2+4dz

|z - 2i| = 1

|x + iy - 2i| = 1

|x + i(y - 2)| = 1

Taking the magnitude of the above, we get:

x2 +(y - 2)2 = 1

This is the equation of a circle with:

Radius (r) = 1

Center = (0, 2)

Poles of f(z) = z² + 4

z = (+2i, -2i)

2i lies Inside the circle

-2i lies outside the circle

∴ We can write:

Forpole(z=−2i)=∫1(z−2i)(z+2i)dz=0

Now, for pole z = 2i (lies inside the circle)

∫1(z+2i)(z−2i)dz=2πi[12i+2i]z=2i