Hyperbolic Functions

Complex Analysis · Axiom Academy

and built from e^z — and the bridge that makes them trig rotated by i . Take e^ z and its mirror e^ -z . Their average is ; half their difference is . Watch the two exponential curves combine: the sum bows upward into , the difference rocks through the origin into . cosh — the sum (even, never below 1) sinh — the difference (odd, through the origin) For a real parameter t , the point never leaves the curve x^2 - y^2 = 1 — the right branch of a hyperbola . Where rides a circle ( x^2 + y^2 = 1 ), the hyperbolic pair rides a hyperbola. One sign flip; a whole different curve. always, so the point stays on the right branch — the curve never crosses the y -axis. runs over all reals, sweeping the point up and down the branch as t changes. The Pythagorean identity is (a plus). Flip one sign to a minus and the circle opens into a hyperbola: . That single minus is the whole difference between circular and hyperbolic. Feed an imaginary input to a circular function and it turns hyperbolic. Rotating the input by i swaps the circle for the hyperbola: of iz is exactly , and of iz is . Hyperbolic and circular functions are one family, rotated by i . The same swap runs the other way: and . Each function on the circle has a hyperbolic twin reached by rotating the input through i . You've seen and built from , the identity that names them, and the rotation by i that ties them to and . Scroll up to revisit any step.

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