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MA 201 Complex Analysis Lecture 6: Elementary Functions

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MA 201 Complex Analysis Lecture 6: Elementary Functions MA 201 Complex Analysis Lecture 6: Elementary functions Lecture 6 Elementary functions The Exponential Function Recall: iy Euler's Formula: For y 2 R; e = cos y + i sin y x+y x y and for any x; y 2 R; e = e e . Definition: If z = x + iy, then ez or exp(z) is defined by the formula ez = e(x+iy) = ex (cos y + i sin y): Image of following sets under exponential function: x f(x; y0): x 2 Rg 7−! f(r; θ): r = e ; x 2 R; θ = y0g: x0 f(x0; y): y 2 Rg 7−! f(r; θ): r = e ; θ 2 Rg: f(x; y): a ≤ x ≤ b; c ≤ y ≤ dg 7−! f(r; θ): ea ≤ r ≤ eb; c ≤ θ ≤ dg: Lecture 6 Elementary functions Properties of Exponential Function e0 = 1 ez = ez¯ ; jez j ≤ ejzj: z z x iy x e 6= 0; for all z 2 C: Look at je j = je jje j = e 6= 0: z+w z w For all z; w 2 C, e = e e : Indeed, if z = x + iy; w = s + it then, ez+w = e(x+s)+i(y+t) = e(x+s)[cos(y + t) + i sin(y + t)] = ex es [(cos y cos t − sin y sin t) + i(sin y cos t + cos y sin t)] = [ex (cos y + i sin y)][es (cos t + i sin t)] = ez ew : Lecture 6 Elementary functions Properties of Exponential function A function f : C ! C is called periodic if there is a w 2 C (called a period) such that f (z + w) = f (z) for all z 2 C: ez is periodic function with period 2πi. ez is not injective unlike real exponential. z x x Since e = e cos y + ie sin y satisfies C-R equation on C and has continuous first order partial derivatives. Therefore ez is an entire function. d z @ x @ x z In fact d z e = @x (e cos y) + i @x (e sin y) = e : Lecture 6 Elementary functions Trigonometric Functions Define 1 1 sin z = (eiz − e−iz ); cos z = (eiz + e−iz ): 2i 2 Properties: sin2 z + cos2 z = 1 sin(z + w) = sin z cos w + cos z sin w cos(z + w) = cos z cos w − sin z sin w sin(−z) = − sin z and cos(−z) = cos z sin(z + 2kπ) = sin z; and cos(z + 2kπ) = cos z 1 sin z = 0 () z = nπ and cos z = 0 () z = (n + 2 )π; ; n 2 Z: Lecture 6 Elementary functions Trigonometric functions sin z; cos z are entire functions. d d dz (sin z) = cos z and dz (cos z) = − sin z: sin z and cos z are unbounded functions. Define: sin z tan z = cos z cos z cot z = sin z 1 sec z = cos z 1 csc z = : sin z Lecture 6 Elementary functions Trigonometric functions Hyperbolic Trigonometric functions: Define ez − e−z ez + e−z sinh z = ; and cosh z = : 2 2 Properties: sinh z; cosh z are entire functions. cosh2 z − sinh2 z = 1. sinh(−z) = − sinh z; cosh(−z) = cosh z; sinh(z + 2kπ) = sinh z; cosh(z + 2kπ) = cosh z; k 2 Z. sinh(iz) = i sin z and cos(iz) = cos z sin z = sin(x + iy) = sin x cosh y + i cos x sinh y; cos z = cos(x + iy) = cos x cosh y − i sin x sinh y. Lecture 6 Elementary functions Complex Logarithm z ∗ Note that e is an onto function from C to C = C n f0g: In fact if ∗ iθ w 2 C then w = jwje where θ 2 (−π; π]: If we set z = log jwj + iθ then, ez = elog jwj+iθ = elog jwjeiθ = w: z z+2πik z e is not an injective function as e = e ; k 2 Z: But if we restrict the domain then it becomes injective. In particular if H = fz = x + iy : −π < y ≤ πg z then z ! e is a bijective function from H to C n f0g: Question: What is the inverse of this function? Lecture 6 Elementary functions Complex Logarithm ∗ Definition: For z 2 C , define log z = ln jzj + i arg z: ln jzj stands for the real logarithm of jzj: Since arg z = Arg z + 2kπ; k 2 Z it follows that log z is not well defined as a function. (multivalued) ∗ For z 2 C , the principal value of the logarithm is defined as Log z = ln jzj + i Argz: ∗ Log : C ! fz = x + iy : −π < y ≤ πg is well defined (single valued). Log z + 2kπi = log z for some k 2 Z: Lecture 6 Elementary functions Complex Logarithm If z 6= 0 then eLog z = eln jzj+i Argz = z What about Log (ez )?. Suppose x is a positive real number then Log x = ln x + i Argx = ln x. π iπ Log i = ln jij + i 2 = 2 ; Log (−1) = ln j − 1j + iπ = iπ; −π iπ Log (−i) = ln j − ij + i 2 = − 2 ; Log (−e) = 1 + iπ Lecture 6 Elementary functions Complex Logarithm The function Log z is not continuous on the negative real axis − R = fz = x + iy : x < 0; y = 0g. To see this consider the point z = −α; α > 0: Consider the sequences i(π− 1 ) i(−π+ 1 ) fan = αe n g and fbn = αe n g: Then lim an = z = lim bn n!1 n!1 but 1 lim Log an = lim ln α + i(π − ) = ln α + iπ n!1 n!1 n and lim Log bn = ln α − iπ: n!1 Lecture 6 Elementary functions Complex Logarithm ∗ − iθ The function z ! Log z is analytic on the set C n R : Let z = re 6= 0 and θ 2 (−π; π): Then Log z = ln r + iθ = u(r; θ) + iv(r; θ) with u(r; θ) = ln r and v(r; θ) = θ: Then 1 1 1 u = v = and v = − u : r r θ r r r θ The identity Log (z1z2) = Logz1 + Log z2 is not always valid. However, the above identity is true if and only if Arg z1 + Arg z2 2 (−π; π] (why?). Lecture 6 Elementary functions Complex Logarithm Branch of a multiple valued function: Let F be a multiple valued function defined on a domain D. A function f is said to be a branch of the multiple valued function F if in a domainD0 ⊂ D if f (z) is single valued and analytic in D0. Branch Cut: The portion of a line or a curve introduced in order to define a branch of a multiple valued function is called branch cut. Branch Point: Any point that is common to all branch cuts is called a branch point. Lecture 6 Elementary functions Complex Exponents Let w 2 C. For any z 6= 0, define z w = exp(w log z); where \exp" is the exponential function and log is the multiple valued logarithmic function. z w is a multiple valued function. i π π i = exp[i log i] = exp[i(log 1 + i 2 )] = exp(− 2 ). Lecture 6 Elementary functions.
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