请输入您要查询的字词:

 

单词 DualityOfGudermannianAndItsInverseFunction
释义

duality of Gudermannian and its inverse function


There are a lot of formulae concerning the Gudermannian functionDlmfPlanetmath and its inverse function containing a hyperbolic functionDlmfMathworldPlanetmath or a trigonometric functionDlmfMathworldPlanetmath or both, such that if we change functionsMathworldPlanetmath of one kind to the corresponding functions of the other kind, then the new formula also is true.

Some exemples:

gdx=0xdtcosht,gd-1x=0xdtcost(1)
ddxgdx=1coshx,ddxgd-1x=1cosx(2)
tan(gdx)=sinhx,tanh(gd-1x)=sinx(3)
sin(gdx)=tanhx,sinh(gd-1x)=tanx(4)
tangdx2=tanhx2,tanhgd-1x2=tanx2(5)

For proving (5) we can check that

ddx[2arctan(tanhx2)]=1coshx,

and since both the expression in the brackets and the http://planetmath.org/node/11997Gudermannian vanish in the origin, we have

gdx 2arctan(tanhx2).

This equation implies (5).

The duality (http://planetmath.org/DualityInMathematics) of the formula pairs may be explained by the equality

gdix=igd-1x.(6)
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 13:03:39