4 ms·
1^x = 1 for any complex x. So that library can be very simple and fast!
by oh_my_goodness 1mo ago
1^x = 1 for any complex x. So that library can be very simple and fast!
- adrian_b 1mo agoWrong. 1^x is multi-valued, and like with all other multi-valued complex functions it is possible to select a branch of the function that is a proper function. Defined correctly, the value of 1^x for any rational x is the corresponding smallest root of unity, and for irrational arguments it is defined by continuity. Therefore 1^x rotates on the unit circle for increasing x. The standard complex exponential and complex logarithm functions are defined exactly in the same way, because they are also multi-valued, so the same kind of equalities like yours would also be true for them. With the correct definition, 1^x = 1 only for integer x, not for any x. As I have written above, the useful 1^x is defined only for real arguments, not for complex arguments. Only its values are complex numbers of unit modulus.
- oh_my_goodness 1mo ago>The standard complex exponential and complex logarithm functions are defined exactly in the same way, because they are also multi-valued, so the same kind of equalities like yours would also be true for them. Example values of e^z and 1^z: e^2 ~= 7.389 1^2 = 1 e^(i\pi) =-1 1^(i\pi) = 1 e^-7 ~= 0.000912 1^(-7) = 1 e^(1+i) ~= 1.46869394 + 2.28735529 i 1^(1+i) = 1