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Elementary Functions > Coth[z] > Introduction to the Hyperbolic Cotangent Function

Definition of the hyperbolic cotangent for a complex argument

In the complex ‐plane, the function is defined by the same formula as for real values.

In the points , where has zeros, the denominator of the last formula equals zero and has singularities (poles of the first order).

Here are two graphics showing the real and imaginary parts of the hyperbolic cotangent function over the complex plane: