Introduction To Hyperbolic Functions 20.pdf
Rosalee Ocegueda
From these three basic functions, the other functions such as hyperbolic cosecant (cosech), hyperbolic secant(sech) and hyperbolic cotangent (coth) functions are derived. Let us discuss the basic hyperbolic functions, graphs, properties, and inverse hyperbolic functions in detail.
The inverse function of hyperbolic functions is known as inverse hyperbolic functions. It is also known as area hyperbolic function. The inverse hyperbolic function provides the hyperbolic angles corresponding to the given value of the hyperbolic function. Those functions are denoted by sinh-1, cosh-1, tanh-1, csch-1, sech-1, and coth-1. The inverse hyperbolic function in complex plane is defined as follows:
Among these, three types of numbers are represented: integers, irrational numbers and imaginary numbers. Three of the basic mathematical operations are also represented: addition, multiplication and exponentiation.
To be sure, these do presuppose properties of exponent such as $e^z_1+z_2=e^z_1 e^z_2$ and $e^-z_1 = \frac1e^z_1$, which for example can be established by expanding the power series of $e^z_1$, $e^-z_1$ and $e^z_2$.
Had we used the rectangular $x + iy$ notation instead, the same division would have required multiplying by the complex conjugate in the numerator and denominator. With the polar coordinates, the situation would have been the same (save perhaps worse).
If anything, the exponential form sure makes it easier to see that multiplying two complex numbers is really the same as multiplying magnitudes and adding angles, and that dividing two complex numbers is really the same as dividing magnitudes and subtracting angles.
For example, by subtracting the $e^-ix$ equation from the $e^ix$ equation, the cosines cancel out and after dividing by $2i$, we get the complex exponential form of the sine function:
For example, by starting with complex sine and complex cosine and plugging in $iz$ (and making use of the facts that $i^2 = -1$ and $1/i = -i$), we have: \beginalign* \sin iz & = \frace^i(iz)-e^-i(iz)2i \\ & = \frace^-z-e^z2i \\ & = i \left(\frace^z-e^-z2\right) \\ & = i \sinh z \endalign* \beginalign* \cos iz & = \frace^i(iz)+e^-i(iz)2 \\ & = \frace^z + e^-z2 \\ & = \cosh z \endalign* From these, we can also plug in $iz$ into complex tangent and get: \[ \tan (iz) = \frac\sin iz\cos iz = \fraci \sinh z\cosh z = i \tanh z \] In short, this means that we can now define hyperbolic functions in terms of trigonometric functions as follows:
But then, because the complex logarithm is now well-defined, we can also define many other things based on it without running into ambiguity. One such example would be the general complex exponential (with a non-zero base $a$), which can be defined as follows:
For example, using the general complex exponential as defined above, we can now get a sense of what $i^i$ actually means: \beginalign* i^i & = e^i \ln i \\ & = e^i \frac\pi2i \\ & = e^-\frac\pi2 \\ & \approx 0.208 \endalign*
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.
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 $e^x$ useful? We now have "mini logarithms" and "mini exponentials", with partial versions of $e$'s famous properties.
I can handle it: how do hyperbolas connect to exponentials? Hyperbolas come from inversions ($xy = 1$ or $y = \frac1x$). The area under an inversion grows logarithmically, and the corresponding coordinates grow exponentially. If we rotate the hyperbola, we rotate the formula to $(x-y)(x+y) = x^2 - y^2 = 1$. The area/coordinates now follow modified logarithms/exponentials: the hyperbolic functions.
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.
Now, why the adjective "hyperbolic"? Euler used the quantity $(e^x + e^-x)$ without giving it a special name. Lambert later called them "transcendental logarithmic functions", and even later the ability to build hyperbolas was seen. That use case has stuck.
Neat. The hyperbolic functions are like "half exponentials" because it takes two derivatives to complete the cycle. This is why they're useful in calculus -- not because we care about the coordinates on a hyperbola!
If we feed the imaginary axis to our everyday exponential function, we get the trig functions which live on a circle. The rotation is happening via the parameter $ix$, vs. in the function $e^ix$. (Instead of rotating the steering wheel, we're rotating the engine, so to speak.)
A short while back, I'd have no idea how to make a parabola gently transition into a line. But seeing $\cosh$ as "parabolic short term, exponential long term" gives us a clue: use the natural log to undo the "exponential long term" behavior, giving us "parabolic short term, linear long term".
Almost there. Our original hyperbola ($xy = 1$) contains the point $(1, 1)$, which is a distance of $\sqrtx^2 + y^2 = \sqrt1^2 + 1^2 = \sqrt2$ from the origin. The constraint equation is really $x^2 - y^2 = r^2$.
The solution turns out to be the even and odd parts of the exponential, $\cosh$ and $\sinh$. There's a 1950's pamphlet "Hyperbolic Functions" by V. G. Shervatot, that goes through the derivation. The key intuition is realizing that hyperbolas (generally speaking) trap area logarithmically, so the necessary coordinates grow exponentially:
It's cute that $\cosh$ parameterizes a hyperbola, but that interpretation has nothing to do with why it's the solution. I think "the catenary follows the even part of the exponential function" not "the catenary follows the x-coordinate of the hyperbola".
For large $x$, the $-1$ is negligible and $\sqrt\textcurrent value^2 - 1 \sim \textcurrent value$. So, for large $x$, we get equality between area, arc length, and current value (imagine the green rope hanging down and just touching the x-axis). It's more connected than regular $e^x$, not bad!
(Intuition for another day: Math deals with unitless quantities. $13 \ \textcm$ is not directly comparable with $13 \ \textcm^2$. Yet in math class, we can solve $x = 1 + x^2$ and nobody cares that constant, linear and squared terms are used in conjunction.)
The shape of the universe may be a hyperbola, and hyperbolic geometry is used in special relativity (beyond my pay grade). If we do live in a giant hyperbola, I, uh, may be forced to recant my "exponentials first" stance.
The hyperbolic functions can be seen as exponential functions (relating time and growth) or geometric functions (relating area and coordinates). Hyperbolas, generally speaking, have logarithmic area and exponential coordinates.
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