Why are hyperbolic functions defined in terms of exponentials?


by neginf
Tags: defined, exponentials, functions, hyperbolic, terms
neginf
neginf is offline
#1
Feb10-11, 12:56 PM
P: 56
Where do the definitions of hyperbolic functions in terms of exponentials come from ?
Phys.Org News Partner Mathematics news on Phys.org
Researchers help Boston Marathon organizers plan for 2014 race
'Math detective' analyzes odds for suspicious lottery wins
Pseudo-mathematics and financial charlatanism
neginf
neginf is offline
#2
Feb10-11, 02:25 PM
P: 56
Maybe somebody tried e^i*x=cos x+i*sin x and e^-i*x=cos x-i*sin x with i*x instead of x and got functions that acted a lot like regular trig functions so they named them a lot like trig functions, like cos(i*x)=cosh x.
Could that be where those definitions come from ?
If so,do they come from somewhere else also ?
Hurkyl
Hurkyl is offline
#3
Feb10-11, 03:07 PM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,101
AFAIK, the hyperbolic trigonometric functions were originally defined geometrically, analogously to the ordinary (circular) trigonometric functions, just based on the rectangular hyperbola rather than the circle.

neginf
neginf is offline
#4
Feb10-11, 03:40 PM
P: 56

Why are hyperbolic functions defined in terms of exponentials?


Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.
AlephZero
AlephZero is offline
#5
Feb10-11, 04:10 PM
Engineering
Sci Advisor
HW Helper
Thanks
P: 6,356
Quote Quote by neginf View Post
Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.
The most "obvious" formula from the geometry of a hyperbola would be cosh^2 x = sinh^2 x + 1.

Another "obvious" starting point from a rectangular hyperbola is to define log(x) as the integral of 1/x, and exp(x) as the inverse of log(x).

From the point of view of "advanced" math, probably the simplest way to define the trig hyperbolic and exponential functions is using power series of complex variables. Then the relations between them are obvious (and they are all so-called analytic functions defined for all complex arguments, which means they have lots of nice properties), but you then have to prove they have something to do with angles, circles, and hyperbolas. But many of the their uses in "advanced" math and physics don't have much to do with angles and hyperbolas anyway...


Register to reply

Related Discussions
Reading a paper, need several terms defined Aerospace Engineering 0
Is there a reason the Ampere is defined in terms of negative charge per unit time? Classical Physics 7
Hyperbolic trigonometric functions in terms of e ? General Math 6
hyperbolic area sector confirmed with hyperbolic functions Calculus 0
Hedonism defined in terms of sentience. General Discussion 2