Hyperbolic Functions For Complex Numbers
Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are then holomorphic.
Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:
so:
Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( for hyperbolic tangent and cotangent).
Read more about this topic: Hyperbolic Function
Famous quotes containing the words functions, complex and/or numbers:
“One of the most highly valued functions of used parents these days is to be the villains of their childrens lives, the people the child blames for any shortcomings or disappointments. But if your identity comes from your parents failings, then you remain forever a member of the child generation, stuck and unable to move on to an adulthood in which you identify yourself in terms of what you do, not what has been done to you.”
—Frank Pittman (20th century)
“In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
—Georg Wilhelm Friedrich Hegel (17701831)
“Old age equalizeswe are aware that what is happening to us has happened to untold numbers from the beginning of time. When we are young we act as if we were the first young people in the world.”
—Eric Hoffer (19021983)