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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 Function. The real logarithm function ln x is defined as the inverse of the exponential function — y = ln x is the unique solution of the equation x = ey. This works because ex is a one-to-one function; if x1 6=x2, then ex1 6=ex2.
Example: Use the definitions of cos(z ) and sin(z ), eiz. + e−iz. cos(z ) = sin(z ) = 2 , to find (cos(z )) and (sin(z )) . Show that Euler’s formula also works if. eiz − e−iz. 2i . θ is complex.
Completing our example of using Cardano's formula Elementary functions of a complex variable. 4.1. Polynomials and rational functions. 4.2. Exponential and logarithm functions. 4.3. General power function. 5. 4.4. Trigonometric and inverse trigonometric functions Metric and topology in complex plane.
Main examples: I = [0; 2 ) or I = ( ; ] : arg[0;2 )( 3 ; i) = 2. arg( ; ]( = argI(z) has a cut line where the value jumps by 2 . Complex logarithms. To solve ew = z , for z 6= 0 : Let w = u + iv ; so ew = eueiv ; and write z = jzj e i : eueiv = jzj e i , eu = jzj ; eiv = ei : Infinitely many solutions w : u = log jzj ; v = + 2 k.
Definition. Let z and a be complex numbers then the power za is defined as a log(z) This is generally multiple-valued, so to specify a single value requires choosing a branch of log(z). 3.5.2 Complex powers
In calculus, interesting examples of differentiable functions, apart from polynomi-als and exponential, are given by trigonometric functions. The situation is similar for functions of complex variables. If x ∈ R then using Taylor series for sine and cosine we get eix = X∞ n=0 (ix)n n! = ∞ n=0 (−1)n x2n (2n)! +i X∞ n=0 (−1)n x2n+1 ...
