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Since f (z) is analytic at 0, f (z) p = å¥ n=0 anzn in some disk jzj < r. Therefore, g(z) = f (z2) = å¥ n=0 anz2n in jzj < r and hence. m=0 m! And since the power series representation of an analytic function is unique, we must have g(m)(0) = 0 for m is odd, i.e., m = 2n 1 for all positive integers n. 4.
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.
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.
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, √.
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 ...
