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That formula as such has no importance at all, it is just a correspondence between different notations. What you really mean is that there are certain mathemat
by adrian_b 2mo ago
That formula as such has no importance at all, it is just a correspondence between different notations.
What you really mean is that there are certain mathematical problems where the complex exponential function is useful and for the complex exponential it is more convenient to measure the real part in nepers and the imaginary part in radians (in order to have a simple formula for computing its derivative and its primitive).
However, the problems where the complex exponential is truly useful are at least an order of magnitude less numerous than it appears from the manner in which mathematics is taught in schools, following a tradition from the 19th century, when symbolic computations done with pen and paper were more important than numeric computations.
For the vast majority of practical problems, the complex exponential is not useful at all (i.e. using it introduces unnecessary complications, without providing any advantage), but a pair of other exponential functions is much more useful, because they ensure computations that are both faster and more accurate: the binary exponential 2^x, with real argument and value, and the exponential 1^x, with argument measured in cycles and a value that is a complex number of unit norm (unit modulus).
Using the pair of exponentials from above, removes computations that are slow and inaccurate (for the reduction of the arguments) from each function evaluation, by moving them to the computation of derivatives or primitives, which are operations that happen much more seldom and which also can frequently be done at compile time, instead of at run time.
Any algorithm that is described by using complex exponentials can be rewritten to use only 2^x and 1^x, and this normally allows various simplifications in the numeric computations. Even the formulae for Fourier transforms are simpler.
The author of the TFA is perfectly right and the use of nepers and radians is a very bad habit, which is a handicap with which most people remain after being taught mathematics in schools, with antiquated methods.
While in TFA the author uses the term "turn" for the unit of plane angle, the traditional name, which was used until some time after WWII, was "cycle", from which various other unit names where derived, e.g. "cycles per meter" or "cycles per second".
- elictronic 2mo agoIt’s good to know you don’t want your house powered by AC power anymore. It’s a weird take, but you do you.
- adrian_b 2mo agoAs I have already said, and I am an electronics engineer, the design of any electrical or electronic system, including anything that uses AC power, can be done faster and with less numerical errors, if instead of using complex exponentials one uses 2 other exponential functions: "2^x" and "1^x", with arguments and value domains as I have written above. Unfortunately, this requires the use of a custom library of transcedental functions, because the standard libraries of most programming languages provide only the functions with arguments in nepers and radians. The standard C library has attempted to add the missing functions, but for a completely impossible to understand reason the standard has defined wrongly the trigonometric functions, requiring arguments measured in half cycles, instead of cycles. This is an extraordinarily stupid mistake, but at least if a standard C library implementation has all the functions, it is easy to modify the sources of the trigonometric functions with arguments in half cycles, to take arguments in cycles instead.
- mike-the-mikado 2mo agoA similar thought (about 1^x) had occurred to me, but I wasn't aware that it was common practice (in some field, at least). Do you have a reference?
- deleted 2mo ago[deleted]
- oh_my_goodness 2mo ago1^x = 1 for any complex x. So that library can be very simple and fast!
- adrian_b 2mo 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 2mo 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