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Reproduced source https://zhuanlan.zhihu.com/p/20042215
First, the development history
Origin hyperbolic function is catenary, who first proposed the catenary shape of the problem is Leonardo da Vinci. He painted "hold silver marten woman" had ponder shape necklace black woman on the neck, unfortunately he did not get the answer she died.
After a lapse of 170 years, the famous Jacob Bernoulli in the paper, also raised the issue and try to prove that this is a parabola. In fact, Galileo before him and Girard have speculated that the chain curve is a parabola.
A year later, Jacob proof of the lack of progress (nonsense, proof of how wrong things will progress). While his brother John Bernoulli has solved the correct answer, Leibniz same period also gives the correct equations of catenary. Their methods are the use of calculus, differential equation given secondary catenary then solved in accordance with the laws of physics.
18 century, John Lambert began to study the function, for the first time introduced the hyperbolic trigonometric functions; in the late 19th century, Augustus De Morgan circle trigonometry extended to hyperbolic, then William Clifford hyperbolic angle parameter of the hyperbola units. So far, the hyperbolic functions in mathematics has occupied a pivotal position.
There is a discipline began a comprehensive development of the 19th century - a complex function. With the birth of Euler's equation, hyperbolic and trigonometric functions of these two types look very different function gained unprecedented unity .
Second, the function definition
Before talking about the definition of hyperbolic functions, let's look at the definition of trigonometric functions. as the picture shows:
In the real domain value by a trigonometric function unit circle and trigonometric angle end edge defining the length of the line . Of course, this "length" is positive or negative.
Similarly, the value of the length is also a hyperbolic function and the hyperbolic angle end edge of the line defined by hyperbolic function . Figure:
Specifically defined as
Third, the nature of the function
Trigonometric functions and the corresponding properties very similar, but there are some differences.
Four, identity
Hyperbolic function identities must be combined with trigonometric identities watching, really too much like:
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