Αποτελέσματα Αναζήτησης
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.
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 argz is determined only to a multiple of 2π, each nonzero complex number has an infinite number of logarithms. For example, log(1+i)=ln √ 2+(π/4+2kπ)i =(1/2)ln2+ (8k +1)πi/4.
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
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
Chapter 13: Complex Numbers. Sections 13.5, 13.6 & 13.7. 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.
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.
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 ...
