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## Complex Variables

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**Complex numbers are really two numbers packaged into one**entity (much like matrices). The two “numbers” are the real and imaginary portions of the complex number:**We may plot complex numbers in a complex plane: the**horizontal axis corresponds to the real part and the vertical axis corresponds to the imaginary part. Im{z} z = x + jy y Re{z} x**Often, we wish to use polar coordinates to specify the**complex number. Instead of horizontal x and vertical y, we have radius r and angle q. Im{z} z = x + jy y r Re{z} q x**The best way to express a complex number in polar**coordinates is to use Euler’s identity: So, and**We also have**A summary of the complex relationships is on the following slide.**Im{z}**r y = r sin q q Re{z} x = r cos q**The magnitude of a complex number is the square-root of the**sum of the squares of the real and imaginary parts: If we set the magnitude of a complex number equal to a constant, we have**or,**This is the equation of a circle, centered at the origin, of radius c.**Im{z}**c2 = |z|2 = x2 + y2 z = x + jy y c Re{z} x**Suppose we wish to find the region corresponding to**This would be a disk, centered at the origin, of radius c.**Im{z}**x2 + y2 = |z|2 < c2 y c Re{z} x**Suppose we wish to find the region corresponding to**This would be a disk, centered at z0, of radius c.**Im{z}**(x-x0)2 + (y-y0)2 = |z-z0|2 < c2 = |z-z0|2 c y0 z0 Re{z} x0**Functions of Complex Variables**Suppose we had a function of a complex variable, say Since z is a complex number, w will be a complex number. Since z has real and imaginary parts, w will have real and imaginary parts.**The standard notation for the real and imaginary parts of z**are x and y respectively. The standard notation for the real and imaginary parts of w are u and v respectively.**where**Both u and v are functions of x and y.**So a complex function of one complex variable is really two**real functions of two real variables.**Exercise: Find u(x,y) and v(x,y) for each of the following**complex functions:**Continuity of Complex Functions**In order to perform operations such as differentiation and integration of complex functions, we must be able to verify of the complex function is continuous. A complex function is said to be continuous at a point z0 if as z approaches z0 (from any direction) then f(z) can be made arbitrarily close to f(z0).**A more mathematical definition of continuity would be for**any e, we can make for some d such that Since we are dealing with complex numbers, the geometric interpretation of this statement is different from that of real numbers.**The region |z-z0| < d defines a disk in the complex plane of**radius d centered about z0. d Im{z} z0 Re{z}**So, if we wish |f(z)-f(z0)| < e we must find a d to make**this so. e Im{w} f(z0) Re{w}**Example: Suppose**Find d such that for**We can do some calculations on a spreadsheet**(continuity.xls). A value of d< 0.1 seems to do it.**A MATLAB plot (by continuity.m) of the previous example is**shown on the following slide.**Differentiation of Complex Functions**How do we take derivatives of complex functions with respect to complex variables? If what is**The differential dz can vary in one of two ways: along the**real axis (dx) or along the imaginary axis (dy). Im{z} y+dy dy y dx Re{z} x x+dx**As z varies in either direction, the derivative must be the**same. x direction y direction So, we must have**These last two conditions**are called the Cauchy-Riemann equations. These equations are the criteria for a complex function to be differentiable (with respect to z = x + jy).**Example: Show that the function**is differentiable Solution: We have shown that**Now that we have determined that this function is**differentiable, the derivative can be found using or**we have**or**We see that the derivative in both cases is**The answer is what we would expect to get if z were treated as a real variable. As it turns out, for most well-behaved complex functions, the derivative can be found by treating z as if it were a real variable.**Example: Show that the function**is not differentiable Solution: We have shown that**Exercise: Is**differentiable?**Definition: A function**is said to be analytic if it is differentiable throughout a region in the complex plane.**Integration of Complex Functions**What happens when we try to take the integral of a complex function along some path in the complex plane?**A complex integral is like a line integral in two**dimensions. The real and the imaginary parts of the integral are nearly identical to classic line integrals.**Example: Integrate**over the real interval z = 0 + j0 to z = 2 + j0. Solution: We have shown that**Since we are integrating along the real (x) axis, all**integrals with respect to dy are zero. In addition y=0. So,**The result is exactly what we would expect to get if we**simply integrated a real variable from 0 to 2.**Example: Integrate**over the imaginary interval z = 0 + j0 to z = 0 + j2. Solution: The integral becomes