Comment by selecsosi
1 month ago
Guessing the original comment hasn't taken complex analysis or has some other oriented view point into geometry that gives them satisfaction but these expressions are one of the most incredible and useful tools in all of mathematics (IMO). Hadn't seen another comment reinforcing this so thank you for dropping these.
Cauchy path integration feels like a cheat code once you fully imbibe it.
Got me through many problems that involves seemingly impossible to memorize identities and re-derivation of complex relations become essentially trivial
Complex exponentials and complex logarithms are useful in some symbolic computations, those involving formulae for derivatives or primitives, and this is indeed the only application where the use of e^x and natural logarithm is worthwhile.
However, whenever your symbolic computation produces a mathematical model that will be used for numeric computations, i.e. in a computer program, it is more efficient to replace all e^x exponentials and natural logarithms with 2^x exponentials and binary logarithms, instead of retaining the complex exponentials and logarithms and evaluating them directly.
At the same time, it is also preferable to replace the trigonometric functions of arguments measured in radians with trigonometric functions of arguments measured in cycles (i.e. functions of 2*Pi*x).
This replacement eliminates the computations needed for argument range reduction that otherwise have to be made at each function evaluation, wasting time and reducing the accuracy of the results.