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Complex Numbers Analytic Functions and Singularities The Cooper Union Department of Electrical Engineering ECE114 Digital Signal Processing Lecture Notes: Complex Analysis September 6, 2011 Complex Numbers z = x + jy = rej where ej = cos + j sin . Note ej2n = 1 iff n is an integer. j conjugate: z = z = x jy = re 2 z = zz, =argument of z = arg (z) Thej j unit circle is z = 1. log z = ln z + j argj j (z) = ln r + j Actually, logj j z is multiple valued: log z = ln r + j + j2n, n= integer. Note the natural logarithm is written ln for real parameters and log for complex. In MATLAB, log, abs and angle return the (complex or real) natural log, magnitude and argument, respectively. If z, w are complex: zw = ew log z. Thus in general zw is multiple valued, unless w = k = integer: zk is single valued. j j(j/2+j2n) /2 2n j Example 1 j = e = e e , so j has many values, all positive real, with values arbitrarily small (n + ) and arbitrarily large1(n ). ! 1 ! 1 z1/N = e(ln r+j+j2n)/N = r1/N ej/N ej2n/N , which yields N distinct values for 0 n N 1. The N values ej2n/N are the distinct N th roots of unity ( is an N th root of unity≤ ≤ if N j2/N = 1), and are equally spaced around the unit circle. The twiddle factor is WN = e is defined in the context of DFT/FFT. It is a primitive N th root of unity, so-called because its powers W k , 0 k N 1, are all the distinct N th roots of unity. N ≤ ≤ j/4 1/3 j/12 j2/3 j2/3 Example 2 8e = 2e 1, e , e Analytic Functions and Singularities A complex-valued function of a complex variable f (z) can be expressed as: f (z) = u (x, y) + jv (x, y) where u, v are each real-valued functions of two real variables, x, y. The derivative f 0 of f is defined as: f (z + z) f (z) f 0 (z) = lim z 0 z ! If f 0 exists at zo, then f is analytic at zo. If f 0 exists at all points within some region, then f is said to be analytic in the region. If f is analytic for all z, f is called entire. For example, polynomials and ez are entire. The existence of the complex derivative f 0 is a much stronger condition than in the case of real functions. Specifically, as z 0 in any way, the limit must always yield the same answer. ! 1 Example 3 z, z , and most general expressions involving conjugate or magnitude are not j j analytic for any z. Let f (z) = z = x jy. First, if z = x purely real: f (z + x) f (z) = x/x +1 x ! But if z = jy is purely imaginary then: f (z + jy) f (z) = jy/jy 1 jy ! Polynomials, ratios of polynomials, exponential and trigonometric functions are analytic except at points where the denominator goes to zero. For example, tan z = sin z/ cos z is not analytic at z = /2 + n, n= integer. If f is analytic in a region, it may be possible to extend f beyond this region so it stays analytic. This is called analytic continuation. The analytic continuation is uniquely determined by the values on the boundary. n 1 Example 4 f (z) = n1=0 z is analytic for z < 1, but the series does not converge for z 1. However, we can use the formula 1/ (1j j z) to perform analytic continuation to all zj =j ≥ 1. That is, f^(z) =P 1/ (1 z) has the property that f^ = f, z < 1, and f^ is defined and 6 j j n analytic for z 1, z = 1 as well as for z < 1. For example, f (2) = n1=0 2 is nonsense j j ≥ 6 ^ 1 j j but the analytic continuation f (2) = 1 2 = 1 is perfectly valid. P f (z) = u (x, y) + jv (x, y) is analytic at z iff u, v satisfy the Cauchy-Riemann (C-R) equations: @u @v @u @v = , = @x @y @y @x Definition 5 singularity: If f is analytic in a region except at an isolated point zo, then f has a singularity at z = zo. Theorem 6 types of singularities: There are only three types of singularities, which can be classified according to the behavior of limz zo f (z) : ! j j 1. removable singularity: limz zo f (z) exists. If f (zo) is set equal to this limit, f becomes analytic there. For example,! f (z) = sin z/z is analytic at z = 0 if we define f (0) = 1, so z = 0 is a removable singularity. 1. pole: limz f (z) = . Then f must blow up with polynomial growth: f (z) = !1 jm j 1 g (z) / (z zo) where g is analytic at zo, g (zo) being nonzero finite. Equivalently: 0, k > m k lim (z zo) f (z) = finite, not 0, k = m z zo 8 ! < blows up, k < m The multiplicity of the pole is m. : 1 Here, “convergence”of a series or integral ALWAYS refers to absolute convergence. 2 2. essential singularity: limz zo f (z) does not exist: it is neither a finite constant nor . For example, e1/z !0 ifjz = jx, x < 0, x 0, and e1/z if z = x, x > 0, 1 1/z ! ! ! 1 x 0, but e = 1 if z = jy. Thus, limz 0 f (z) does not exist. ! ! j j Theorem 7 Behavior at an essential singularity (Picard’s theorem): If zo is an essential singularity, then in any arbitrarily small neighborhood of zo, f takes on all complex values infinitely often, except for exactly one value. Example 8 f (z) = e1/z has an essential singularity at z = 0. For all > 0 and all w = 0, we can find z < such that f (z) = w. Setting e1/z = w, w = 0, we get: 6 j j 6 1 z = 1/ log w = ln w + j + j2n j j By picking n large enough we can find a z with arbitrarily small magnitude. Note e1/z = 0 for any z. j j 6 To repeat, these are the only possibilities! If f is analytic in a region except for poles, m it is called homomorphic. If f (z) = (z zo) g (z) where g is analytic at zo and g (zo) = 0, 6 then f has a zero at zo of multiplicity m. That is: 0, k < m f (z) lim = finite, not 0, k = m z z k o (z zo) 8 ! < blows up, k > m Similar to the concept of multiplicity of: poles is the multiplicity of zeros. Problem 9 Theorem 10 zero of an analytic function: If f is analytic at z0 and m f (z0) = 0, then either f (z) is identically zero everywhere, or f (z) = (z z0) g (z) where g is analytic at z0 and g (z0) = 0; m is called the multiplicity of the zero. 6 In complex analysis, is considered as a single point. The expression z means that z , the behavior1 of the arg z can be arbitrary. We say f is analytic! at 1 , has polesj orj ! zeros 1 at , etc., according to its behavior as z . Specifically: 1 1 j j ! 1 Theorem 11 behavior at : We say f (z) is analytic at if f (z) is analytic for all 1 1 finite z > R, for some finite radius R, and limz f (z) exists (in this case we denote the limit asj jf ( )). The function f has a zero at !1with multiplicity m if it is analytic there and f (z) =1g (z) /zm where g is analytic at 1and g ( ) = 0; if f ( ) = 0, then either there is finite multiplicity or f is identically zero.1 The1 function6 f has1 a pole at if it is 1 analytic for all finite z > R, for some finite radius R, and limz f (z) = ; in this case, f (z) = zmg (z) jwherej g is analytic at infinity and g ( ) =!1 0,j and mj is1 called the 1 6 multiplicity. If f is analytic for all z > R for some finite R but limz f (z) does not exist and is not , then f has an essentialj j singularity at , and it satisfies!1 j thej behavior described in Picard’stheorem1 (i.e., it takes on all finite complex1 values except exactly one in every region z > R). j j 3 Example 12 Consider: z3 + 1 z3 + 1 z3 + 1 f (z) = ez, f (z) = , f (z) = , f (z) = 1 2 2z + 3 3 2z3 + 3 4 2z5 + 3 Then f1 has an essential singularity at (f1 is still called entire); f2 has a double pole at 2 2 1 (it blows up like z ; f2 (z) /z 1/2 as z ); f3 and f4 are both analytic at , with 1 ! ! 12 2 1 f4 having a double zero there (it dies out like 1/z ; z f4 (z) 1/2 as z ). ! ! 1 A ratio of polynomials is a rational function and is uniquely determined by its poles, zeros and a constant gain factor: n anz + + a1z + a0 (z z1) (z zn) f (z) = m ··· = K ··· bmz + + b1z + b0 (z p1) (z pm) ··· ··· If we count poles or zeros at (corresponding to the cases n > m or n < m, respectively), and count multiplicities, then1 the total number of poles = the total number of zeros = max (n, m).
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