The value of the integral

evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole z = i, where 𝑖 is the imaginary unit, is

Concept:

Residue theorem: if f(z) is an analytic function in a closed curve C except at a finite number of singular points within C then 

•  × (sum of the residues at the singular point within curve C)

Residue for simple pole z = a:

• Res f(a) = 

Calculation:

Given:

pole z = i

Check for singularity at pole z = i

f(z) = 2z⁴ - 3z³ + 7z² - 3z + 5

f(i) = 2(i)⁴ - 3(i)³ + 7(i)² - 3i + 5 

f(i) = 2 ×1 - 3(-i) - 7 - 3i + 5 = 0

since, f(i) = 0 ⇒ z = i  is a singular point

From Residue theorem:

 = 2πi (Residue at z = i )

Residue at z = i :

Res|_{z= i}  = 

at z = i , Res = 0/0 form, applying L'hospital rule 

Res|z= i  =  = 

Res|z= i  =  = 

 = 2πi × 

⇒   = 

 =  = = π(-1 + i)