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Complex Analysis Complex Analysis Prof. Broaddus February 11, 2015 1 More complex functions 1.1 The exponential function Most of the non-rational functions that we will define are based on the exponential function f(z) = ez. Here we list key properties: Proposition 1.1 (Properties of the exponential function). 1. ex+iy = ex cos y + iex sin y (definition) 2. ez+w = ezew 3. ez+2πi = ez (ez is periodic with period 2πi) 4. ez = ez 5. jezj = eRe z 6. There is no z 2 C with ez = 0. 7. For any w 6= 0 there are infinitely many z 2 C with ez = w. z d z z 8. f(z) = e is holomorphic for all z 2 C and dz e = e . We have already proved most of these properties. Proof of Proposition 1.1 Part 4. Compute ez in terms of ez: Let z = x + iy. Then ez = ex cos(−y) + iex sin(−y) = ex cos(y) − iex sin(y) = ex cos(y) + iex sin(y) = ez 1.2 The complex logarithm The real function f(x) = ex is one-to-one so its inverse ln x is a single-valued function. The complex exponential function is 1-to-one so its inverse is an 1-valued function. Definition 1.2 (Complex logarithm). log z = ln jzj + i arg z Notice that this is an 1-valued function. 1 Definition 1.3 (Principal value of the complex logarithm). Log z = ln jzj + i Arg z This is a single-valued function. Most of the non-rational functions that we will define are based on the exponential function f(z) = ez. Here we list key properties: Proposition 1.4 (Properties of the complex logarithm). 1. elog z = eLog z = z if z 6= 0 2. log(zw) = log z + log w if z 6= 0 and w 6= 0 3. log z = Log z + 2πin if z 6= 0 4. log z = log z if z 6= 0 5. Log z = Log z if z2 = R≤0 6. log(ez) = z + 2πin. 7. Log(ez) = z if −π < Im z ≤ π. 8. f(z) = Log(z) is holomorphic for all z 2 C − R≤0 and d 1 Log z = dz z 9. For any choice of branch for log(z) containing a neighborhood of the point z 2 C − f0g d 1 log z = dz z Proof of Proposition 1.4. log z ln jzj+i arg z ln jzj i arg z z 1. e = e = e e = jzj · jzj = z 3. log zw = ln jzwj + i arg zw = ln jzjjwj + i(arg z + arg w) = ln jzj + ln jwj + i arg z + i arg w = log z + log w 4. log z = ln jzj + i arg z = ln jzj − i arg z = log z 6. log ez = ln jezj + i arg ez = ln eRe z + i arg eRe zei Im z = Re z + i(Im z + 2πn) = z + 2πin p 2 2 y 8. Suppose z = x + iy and x; y > 0. Then Log z = ln jzj + i Arg z = ln x + y + i arctan x = 1 2 2 y 2 ln(x + y ) + i arctan x x 1=x x ux = x2+y2 ; vy = 1+(y=x)2 = x2+y2 y −y=x2 −y uy = x2+y2 ; vx = 1+(y=x)2 = x2+y2 Thus Log z is holomorphic and d x iy 1 dz Log z = ux + ivx = x2+y2 − x2+y2 = z : In defining Arg z we started with arg z and chose a particular value for each z so that the resulting function is continuous (in fact holomorphic) except on R≤0 2 Definition 1.5 (Branch). A branch of a multiple-valued function f is a single-valued holomorphic function F on a connected open set where F (z) is one of the values of f(z). A branch cut is a line or curve used to delineate the domain for a particular branch. Example 1.6. The following are all branches of f(z) = log z: 1. F (z) = ln jzj + i Arg z 2. G(z) = ln jzj + i Arg z + 14πi π π 3. H(z) = ln jzj + ia(z) where a(z) is value of arg z in [ 3 ; 3 + 2π). 3.
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