Do Residues and Singularities Define Functions Like tanh(z) and tan(z)?

[pivoxa15]

Homework Statement

Do residues exist only for holomorphic function?

Classify the singularity and calculate the residue of tanh(z) and tan(z)

The Attempt at a Solution

For both
Essential isolated singularity because the numerator has an infinite number of terms.

Residue = 0.

 

[HallsofIvy]

Homework Statement

Do residues exist only for holomorphic function?

Classify the singularity and calculate the residue of tanh(z) and tan(z)

The Attempt at a Solution

For both
Essential isolated singularity because the numerator has an infinite number of terms.

Residue = 0.

What are the definitions of "residue" and "holomorphic function"?

At what point do tanh(z) and tan(z) have an isolated singularity? It doesn't make sense to talk about the "residue" of a function. A function may have a "residue" at a specific point.