Standard Algebraic Expressions
(b) sinh(x) is half the difference of ex and e−x Hyperbolic functions (a) cosh and (b) sinh obtained using exponential functions andThe hyperbolic functions are:
- Hyperbolic sine:
- Hyperbolic cosine:
- Hyperbolic tangent:
- Hyperbolic cotangent:
- Hyperbolic secant:
- Hyperbolic cosecant:
Hyperbolic functions can be introduced via imaginary circular angles:
- Hyperbolic sine:
- Hyperbolic cosine:
- Hyperbolic tangent:
- Hyperbolic cotangent:
- Hyperbolic secant:
- Hyperbolic cosecant:
where i is the imaginary unit defined by i2 = −1.
The complex forms in the definitions above derive from Euler's formula.
Note that, by convention, sinh2 x means (sinh x)2, not sinh(sinh x), and similarly for the other hyperbolic functions when used with positive exponents. In contrast, sinh−1 x refers to the inverse function arsinh x and not to a reciprocal (and again likewise for the other hyperbolic functions).
Read more about this topic: Hyperbolic Function
Famous quotes containing the words standard, algebraic and/or expressions:
“There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
—Blaise Pascal (16231662)
“I have no scheme about it,no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (18171862)
“Many expressions in the New Testament come naturally to the lips of all Protestants, and it furnishes the most pregnant and practical texts. There is no harmless dreaming, no wise speculation in it, but everywhere a substratum of good sense. It never reflects, but it repents. There is no poetry in it, we may say, nothing regarded in the light of beauty merely, but moral truth is its object. All mortals are convicted by its conscience.”
—Henry David Thoreau (18171862)