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  1. z = ia is a simple pole and  is a residue at z = ia of f(z)
  2. z = ia is a simple pole ia is a residue at z = ia of f(z)
  3. z = ia is a simple pole and  is a residue at z = ia of f(z)
  4. none of the above

Answer (Detailed Solution Below)

Option 1 : z = ia is a simple pole and  is a residue at z = ia of f(z)

Detailed Solution

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Concept:

Pole:

The value for which f(z) fails to exists i.e. the value at which the denominator of the function f(z) = 0.

When the order of a pole is 1, it is known as a simple pole.

Residue:

If f(z) has a simple pole at z = a, then

If f(z) has a pole of order n at z = a, then

Calculate:

Given:

For calculating pole:

z2 + a2 = 0

∴ (z + ia)(z - ai) = 0

∴ z = ai, -ai.

∴ z has simple pole at z = ai and -ai.

Residue:

If f(z) has a simple pole at z = a, then

For pole at z = ai

For pole at z = -ai

∴ z has a simple pole at z = ai and  is a residue at z = ia of f(z)

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