Intuitive Guide to Hyperbolic Functions

If the exponential function  is water, the hyperbolic functions ( and ) are hydrogen and oxygen. They’re the technical, rarely-discussed parts that combine into a famous whole.

Admittedly, the hyperbolic functions were tucked into a dark part of my attic. They were defined with strained motivations (“Need yet another way to build a hyperbola?”) then crammed into tables of integrals, soon to be forgotten. I couldn’t think with them.

After much struggle, I found their purpose:

• What are the hyperbolic functions ( and )? The even/odd parts of the exponential function () that, funny enough, can build a hyperbola.
• Why are parts of the exponential called hyperbolic? That’s the modern name. These functions are so darn good at making hyperbolas that they’re typecast for that role. (Similarly, sine isn’t just about circles, and we shouldn’t name it “circular sine”!)
• Why are hyperbolic functions useful? A better framing is: Why are parts of  useful? We now have “mini logarithms” and “mini exponentials”, with partial versions of ‘s famous properties.
• I can handle it: how do hyperbolas connect to exponentials? Hyperbolas come from inversions ( or ). The area under an inversion grows logarithmically, and the corresponding coordinates grow exponentially. If we rotate the hyperbola, we rotate the formula to . The area/coordinates now follow modified logarithms/exponentials: the hyperbolic functions.
• Actually, I couldn’t handle it. That’s ok. We’ll build up to it. These functions took many years to be discovered, and their behavior is hardly obvious.

This post is fairly technical: we’re studying hydrogen, not water. If hyperbolic functions appear in class, you don’t have much choice, and may as well get an intuition. If you’re studying for fun, don’t sweat the details, that’s what calculus students are for.