Validity of orbital element

[Mark Brown <markbrown@xxxxxxxxxxxxx>]

Below I offer mathematical proof that Orbital elements do have
influences upon the financial markets and even Sardine migration.

Hyperbolic trigonometric functions are defined in terms of the natural
exponential function ex.

sinh(x):=[ex-e-x]/2 
cosh(x):=[ex+e-x]/2 
tanh(x):=sinh(x)/cosh(x)=[ex-e-x]/[ex+e-x] 
coth(x):=cosh(x)/sinh(x)=[ex+e-x]/[ex-e-x] 
sech(x):=1/cosh(x)=2/[ex+e-x] 
csch(x):=1/sinh(x)=2/[ex-e-x]

Observe that sinh(x) and cosh(x) are the even and odd components of
ex, by definition.

The following equations relating sinh(x), cosh(x), and ex are special
instances of equations relating even and odd parts of functions to the
function itself:

sinh(x)+cosh(x)=ex
sinh(x)-cosh(x)=e-x

Notation for powers and inverses of hyperbolic trigonometric functions
is similar to that of trigonometric functions:
 
cosh(cosh(x))!=cosh2(x)=cosh(x).cosh(x), but 
1/cosh(x)!=cosh-1(x), and cosh-1(cosh(x))=1 for all real numbers x.

Hyperbolic Pythagorean

cosh2(x)-sinh2(x)=1 
1-tanh2(x)=sech2(x) 
coth2(x)-1=csch2(x)



-- 
Thank You,
Mark Brown
www.markbrown.com