1、1Complex Variable We here go through the complex algebra briefly.A complex number z = (x,y) = x + iy, Where. We will see that the ordering of two real numbers (x,y) is significant, i.e. in general x + iy y + ixX: the real part, labeled by Re(z); y: the imaginary part, labeled by Im(z)Frequently used
2、 representations:(1) Cartesian representation: x+iy(2) polar representation, we may write z=r(cos + i sin) orr the modulus or magnitude of z - the argument or phase of z1iierz2The relation between Cartesian and polar representation:The choice of polar representation or Cartesian representation is a
3、matter of convenience. Addition and subtraction of complex variables are easier in the Cartesian representation. Multiplication, division, powers, roots are easier to handle in polar form, 1 / 2221ta n/rzxyyx212121ierrzz212121/ierrzzinnnerz)()()()(1221212121212121yxyxiyyxxzzyyixxzz3Complex Conjugati
4、on: replacing i by i, which is denoted by (*), We then haveHenceNote: ln z is a multi-valued function. To avoid ambiguity, we usually set n=0and limit the phase to an interval of length of 2. The value of lnz withn=0 is called the principal value of lnz.iyxz*222*ryxzz21*zzz irez nire2 irlnzln nirz2l
5、nln4From z, complex functions f(z) may be constructed. They can be written f(z) = u(x,y) + iv(x,y)in which v and u are real functions. For example if , we have 2121zzzz xyiyxzf222Using the polar form,)arg()arg()arg(2121zzzz2)(zzfDerivatives Differentiation of complex-valued functions is completely a
6、nalogous to the real case: Definition. Derivative. Let f(z) be a complex-valued function defined in a neighborhood of z0. Then the derivative of f(z) at z0 is given byProvided this limit exists. F(z) is said to be differentiable at z0.zzfzzfzfz)()(l i m)(0000Properties of Derivatives R ul e. C hai n.0i f,. const ant anyf or 000020000000000000000zgzgfzgfdzdzgzgzgzfzfzgzgfzgzfzgzfzf gczcfzcfzgzfzgfIntegration
