sympyで「双曲線関数(hyperbolic function)」をやってみた

(参考)simplifying expressions with hyperbolic functions
https://stackoverflow.com/questions/11076263/simplifying-expressions-with-hyperbolic-functions
(参考)チュートリアルを読む 読書ノート v1.5dev ＞ Trigonometric Simplification
https://showa-yojyo.github.io/notebook/python-sympy/tutorial.html#trigonometric-simplification

日本版wikipediaより

(参考)双曲線関数
https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A%E9%96%A2%E6%95%B0#定義

途中から怪しくなります?

fullscript.py

from sympy import *
var('x')
print('定義--------------------------------')
expr=sinh(x)
print(expr,"=",expr.rewrite(exp))
expr=exp(x)-sinh(x)
print(simplify(expr),"=",expr)
expr=tanh(x)
print(expr,"=",simplify(expr*cosh(x)),"/cosh(x)")
expr=coth(x)
print(expr,"=",simplify(expr*tanh(x)),"/tanh(x)")
# expr=sech(x)
# print(expr,"=",simplify(expr*cosh(x)),"/cosh(x)")
# expr=cosech(x)
# print(expr,"=",simplify(expr*sinh(x)),"/sinh(x)")

# 定義--------------------------------
# sinh(x) = exp(x)/2 - exp(-x)/2
# cosh(x) = exp(x) - sinh(x)
# tanh(x) = sinh(x) /cosh(x)
# coth(x) = 1 /tanh(x)
# NotImplementedError: unhandled sech
# NameError: name 'cosech' is not defined

本家wikipediaより

(参考)Hyperbolic function
https://en.m.wikipedia.org/wiki/Hyperbolic_function#Definitions

本家wikipediaよりcosh(x)+sinh(x)

(2018/08/23)
(参考)Relationship to the exponential function
https://en.m.wikipedia.org/wiki/Hyperbolic_function#Relationship_to_the_exponential_function
(参考)simplifying expressions with hyperbolic functions
https://stackoverflow.com/questions/11076263/simplifying-expressions-with-hyperbolic-functions

fullscript.py

from sympy import *
x = Symbol('x')
print((cosh(x)+sinh(x)).simplify(),'=cosh(x)+sinh(x)')
print('-------------------------')
ans=(exp(x)-sinh(x))
print(ans)
ans=(exp(x)-sinh(x)).simplify()
print(ans)
ans=(cosh(x)).rewrite(exp)
print(ans)
ans=(exp(x)-sinh(x)).rewrite(exp)
print(ans)
# exp(x) =cosh(x)+sinh(x)
# exp(x) - sinh(x)
# cosh(x)
# exp(x)/2 + exp(-x)/2
# exp(x)/2 + exp(-x)/2