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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.
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.
8. Prove the following statements. (1) logp b x = 2log x (2) log p1 b p x = log x (3) log 4 x2 = log p x 9. Given that log2 = x, log3 = y and log7 = z, express the following expressions
Complex numbers - Exercises with detailed solutions 1. Compute real and imaginary part of z = i¡4 2i¡3: 2. Compute the absolute value and the conjugate of z = (1+ i)6; w = i17: 3. Write in the \algebraic" form (a+ib) the following complex numbers z = i5 +i+1; w = (3+3i)8: 4. Write in the \trigonometric" form (‰(cosµ +isinµ)) the following ...
16 Νοε 2022 · Here is a set of practice problems to accompany the Logarithm Functions section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University.
Complex logarithm Complex power function Definition Properties 1. Complex exponential The exponential of a complex number z = x +iy is defined 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 entire.
