Answer to Question #16394 in Trigonometry for soni sharma

Question

plz explain me all about Logarithms of complex Quantities .

Answer 1

In complex analysis, a complex logarithm function is an "inverse" of the complex exponential function, just as the natural logarithm ln(x) is the inverse of the real exponential function e^x. Thus, a logarithm of z is a complex number w such that e^w = z. The notation for such a w is ln(z). But because every nonzero complex number z has infinitely many logarithms, care is required to give this notation an unambiguous meaning.
If z = r*e^(i*θ) with r > 0 (polar form), then w = ln(r) + i*θ is one logarithm of z; adding integer multiples of 2πi gives all the others.