Evaluation of "acosh()" function in ADS
I am wondering about the way the function "acosh()" being evaluated in ADS. As I defined an equation namely "Gx = acosh(-0.95 + j0.06)" in a schematic using MeasEqn, after running and plotting in DDS window, this resulted in Gx = -0.17 - j2.78.
However, when I plotted Gx in Mathematica, Mathcad and MATLAB, the results were the same as 0.17 + j2.78. Thus, I am wondering about the result obtained from the calculation in ADS.
Is the "acosh()" function in ADS defined properly ?
Thank you so much guys
DYL
Consider a definition of cosh(Gx).
cosh(Gx) is defined as {exp(Gx)+exp(-Gx)}/2.
For both Gx=-0.17 - j*2.78 and Gx=+0.17 + j*2.78, cosh(Gx)=-0.95 + j*0.06.
ADS defines acosh(z) as log(z+sqrt(z*z-1))
Others define acosh(z) as log(z-sqrt(z*z-1))
However Gx=+0.17 + j*2.78 is more usual than Gx=-0.17 - j*2.78.
Hi pancho_hideboo,
Thank you very much for your explanations. I think that is because acosh() is a double-valued function in nature. So, the sign can be both negative and positive. In most math softwares, they may represent acosh() using the principal value, that's why the results are different than that from ADS.
I am not sure that if the most recent version of ADS still implements acosh(z) using log(z+sqrt(z*z-1)). Perhaps using acosh(z) = ln(z + sqrt(z+1)*sqrt(z-1)) is more general.
Thank you very much again for your answer,
DYL
Not correct.
Rigorously it is multi-valued function not only double-valued.
There is no critical problem at all for me even if Agilent ADS defines acosh(z) like that.
I can't understand why you say this is more general.
Can you understand a multi-valued function correctly ?
"ln(z)" itself is a multi-valued function.