Meaning imaginary result of asin(x) function of MATLAB

Dear all,

I have a question about asin(x) function of MATLAB. As we know from the mathematics, if the argument of arcsin(x) function (e.g. x) is bigger than one, then this function MUST not have any answer. For example, if we calculate arcsin(1.5) with calculator, there is error in the calculator which is obvious and completely clear, but when we write in MATLAB: asin(1.5) it gives us an imaginary result equal to 1.5708 - 0.9624i

My question is how can we interpret this imaginary number? Does it have any meaning? Why does MATLAB give us imaginary answer, while we expect to see error answer by MATLAB?

Before all, I want to thank you for helping me to interpret this imaginary numbers by MATLAB if they have any meaning.

It would be my pleasure, if you can provide me any referrence for your answer also.

I am looking forward to hearing from you. Thank you so much for your time and considerations.

Best regards,

Hadi

[David Wilson]

One thing you could try is sin(asin(1.5)) and see what that equals.

You also might like to look at the analytical continuation of the arcsine function.

https://dlmf.nist.gov/4.23

[Hadi Naderiallaf]

Thank you David. One more question, do absolute value and phase of the complex result of asin function have any especial meaning?

[John D'Errico]

"As we know from mathematics" - this may just indicate one needs to know more mathematics. In this case, it is about the extension of the trig functions to the complex plane.

[Hadi Naderiallaf]

Actually, I have one physical model which have asin function. The output of this model should be a real number not imaginary. Thus, I want to interpret the imaginary output of this physical model and to find if the real part or the absolute value of this imaginary number makes sense in my physical model.

Answer 1

[Stephen23]

"'My question is how can we interpret this imaginary number? Does it have any meaning?

Yes, it has a meaning.

Why does MATLAB give us imaginary answer, while we expect to see error answer by MATLAB?"

I don't expect to get an error, because as we all know, trigonometric functions can be easily extended to the complex domain:

https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Extension_to_complex_plane

https://en.wikipedia.org/wiki/Sine#Relationship_to_complex_numbers

All your example shows is that sin(1.5708 - 0.96242i) == 1.5

If you really want, you can easily calculate this yourself, using the various identities of trigonometric functions on the complex plane:

>> y = 1.5;
>> x = -i*log(y*i+sqrt(1-y^2)) % == arcsine
x =  1.57080 - 0.96242i

[Hadi Naderiallaf]

Thank you Stephen. One more question, do absolute value and phase of the complex result of asin function have any especial meaning?

[John D'Errico]

Special meaning? They mean exactly what the absolute value and phase of any complex number mean. What would you think is special?