Types of Singularities of a Complex Function Theory and Examples GATE

Types of Singularities of a Complex Function Theory and Examples ( GATE )

If a complex function fails to be analytic at a point then it is called singular at that point. The main types of singularities are as follows
1. Isolated singularity: If z = a is singularity of f(z) such that there is no other singularity of f(z) in the neighbourhood of a (an open disc centered at z = a) then the singularity is called an isolated singularity.

2. Removable Singularity: If denominator of a function is zero at z = a but there are no negative powered terms in Laurent series expansion of f(z) about z =a then the singularity is known as removable singularity. The singularity can be removed by redefining the function at z = a.

3. Essential Singularity: If there are infinitely many negative powered terms in the Laurent Series expansion of f(z) about z = a, then it is call essential singularity of f(z).

4. Pole: If the number of negative powers in the Laurent series expansion is finite then the singularity is call a Pole. If the highest negative power is -n then the singularity is called pole of order n. If n = 1, then it is called a simple pole.

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