Αποτελέσματα Αναζήτησης
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.
We use ln only for logarithms of real numbers; log denotes logarithms of com-plex numbers using base e (and no other base is used). Because equation 3.21 yields logarithms of every nonzero complex number, we have defined the complex logarithm function.
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.
The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex Numbers, Functions, Complex Integrals and Series. The majority of problems are provided with answers, detailed procedures and hints (sometimes incomplete solutions).
COMPLEX ANALYSIS: LECTURE NOTES DMITRI ZAITSEV Contents 1. The origin of complex numbers 4 1.1. Solving quadratic equation 4 1.2. Cubic equation and Cardano’s formula 4 1.3. Example of using Cardano’s formula 5 2. Algebraic operations for complex numbers 5 2.1. Addition and multiplication 5 2.2. The complex conjugate 6 2.3. Division 6 3 ...
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 logarithms To solve ew = z , for z 6= 0: Let w = u +iv ;so ew = eueiv;and write z = jzjei : eueiv = jzjei , eu = jzj; eiv = ei : Infinitely many solutions w : u = logjzj; v = +2ˇk Describe all solutions by: log(z) = logjzj+i arg(z) Examples: log(2i) = log(2)+iˇ 2 +i2ˇk log( 3) = log(3)+iˇ+i2ˇk log(2 +5i) = log p 29 +i arctan(5 2 ...
