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Because equation 3.21 yields logarithms of every nonzero complex number, we have defined the complex logarithm function. It is defined for all z 6= 0, and because arg z is determined only to a multiple of 2π, each nonzero complex number has an infinite number of logarithms. For example, √.
The problems covered include finding the general value of log(1+i)+log(1-i), evaluating logarithms of expressions involving trigonometric functions, and using properties of logarithms to simplify complex logarithmic expressions.
8(1 z2)3z; (c) 1=(2z + 1)2; (d) e1=z=z2. 8. Prove the following version of complex L’Hospital: Let f (z) and g(z) be two complex functions defined on jz z0j < r for some r > 0. Suppose that f (z0) = g(z0) = 0, f (z) and g(z) are differentiable at z0 and g0(z0) 6= 0. Then.
This document contains several unsolved problems involving logarithms of complex numbers. The problems involve proving various identities related to taking logarithms and separating results into real and imaginary parts for expressions involving complex numbers and trigonometric functions.
Complex logarithm Complex power function Definition Principal value of ln(|z|) Properties of the logarithm You have to be careful when you use identities like ln(z 1z 2) = ln(z 1)+ln(z 2), or ln z 1 z 2 =ln(z 1)−ln(z 2). They are only true up to multiples of 2πi. For instance, if z 1 = i = exp(iπ/2) and z 2 = −1=exp(iπ), ln(z 1)=i π 2 ...
In these notes, we examine the logarithm, exponential and power functions, where the arguments∗ of these functions can be complex numbers. In particular, we are interested in how their properties differ from the properties of the corresponding real-valued functions.† 1. Review of the properties of the argument of a complex number
Complex logarithm Complex power function De–nition Properties 1. Complex exponential The exponential of a complex number z = x +iy is de–ned as exp(z) = exp(x +iy) = exp(x)exp(iy) = exp(x)(cos(y)+i sin(y)): As for real numbers, the exponential function is equal to its derivative, i.e. d dz exp(z) = exp(z): (1) The exponential is therefore ...
