Postscript or pdf produced by some word processors for output. The complex exponential the exponential function is a basic building block for solutions of odes. In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. Complex numbers and the complex exponential university of. To check this replace the exponentials by their definition. These link the exponential function and the trigonometric functions. The complex logarithm, exponential and power functions. The range of the complex exponential function is the entire complex plane except the zero value. To extract the real and imaginary parts of a given complex number one. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. Two important results in complex number theory are known as eulers relations. In so doing we will come across analytic functions, which form the centerpiece of this part of the course.
Recall that the realvalued functions cosx, sinx, and ex have. Magic with complex exponentials 103 now x t is the sum of a complex number and its complex conjugate. Exponential and trigonometric functions from the first principles. The complex exponential function is defined by extending the taylor series of ex from real values. But when we add a complex number to its complex conjugate, we cancel the imaginary part and double the real part.
Conversely, the sine and cosine functions can be expressed in terms of complex exponentials. This is true also for complex or imaginary numbers. Recall that the maclaurin series of a function is the taylor expansion of the function around zero. In fact, to a large extent complex analysis is the study of analytic.