Now, before you start rolling your eyes and muttering “I already know this stuff,” let me just say: hold on tight because it gets wild.

To start, what are hyperbolic functions? Well, they’re basically the opposite of trigonometric functions instead of dealing with angles and sines/cosines, we’re working with exponentials and logarithms. And for complex numbers, that means we get to play around with some seriously mind-bending formulas.

So let’s dive right in! Here are the four main hyperbolic functions: sinh(z), cosh(z), tanh(z), and sech(z). And just like their trig counterparts, they all have some pretty cool properties that make them useful for solving various equations.

First up is sinh(z) this function takes a complex number z (which can be real or imaginary) and returns another complex number that represents the sine of an angle in hyperbolic space. And just like with regular sines, we can use it to solve all sorts of problems involving exponential growth/decay over time.

Next is cosh(z), which gives us the cosine of a hyperbolic angle. This function is particularly useful for solving equations that involve oscillations or waves in complex space, and can be used to model everything from sound waves to electromagnetic fields.

Then we have tanh(z) this one’s kind of like the tangent of a hyperbolic angle, but with some important differences. For starters, it only works for angles that are less than or equal to pi/2 (since beyond that point, the function becomes undefined). And secondly, unlike regular tangents which can take on any value between -1 and 1, tanh(z) is always bounded by those same limits.

Finally we have sech(z), which gives us the reciprocal of cosh(z). This function is particularly useful for solving equations that involve exponential decay over time (since it can help us find the “half-life” or other key parameters). And like with all hyperbolic functions, it’s also incredibly versatile and can be used to model everything from sound waves to electromagnetic fields.

Whether you’re working on physics problems or just trying to wrap your head around this crazy math stuff, these formulas are sure to come in handy at some point. And who knows? Maybe one day we’ll even be able to use them to solve real-world problems like climate change or space exploration.