Introduction to Complex Variables Class Notes Instructor: Louis Block

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1 Introuction to omplex Vribles lss Notes Instructor: Louis Block

2 Definition 1. (n remrk) We consier the complex plne consisting of ll z = (x, y) = x + iy, where x n y re rel. We write x = Rez (the rel prt of z) n y = Imz (the imginry prt of z). We efine ition n multipliction so tht the xioms of fiel re stisfie. The complex number i stisfies i 2 = 1. Note tht ny rel number x is lso complex number x = x + i0. Note lso tht for ny complex number z, we hve 0 z = 0. Definition 2. Let z = x + iy where x, y re rel. The complex conjugte of z is given by z = x iy. The bsolute vlue of z is given by z = x 2 + y 2. Proposition 3. Let z n w be complex numbers. 1. z + w = z + w. 2. z w = z w. 3. z w = z w. 4. ( z w ) = z w. 5. z z = z z + w z + w. 7. z w = z w. 8. ( z z w ) = w. Remrk 4. Let z n w be complex numbers. Then z w is the usul istnce between z n w consiere s points in the plne. Definition 5. (n remrk) For ny rel number θ we set e iθ = exp(iθ) = cos θ + i sin θ. Then ny complex number z cn be written s z = re iθ, where r = z. Any such θ is clle n rgument of z. The set rguments of z is enote by rg(z). The principl rgument of z, enote by Arg(z) is the unique θ in rg(z) with π < θ π. Proposition 6. Let z 1 = r 1 e iθ1 z 2 = r 2 e iθ2 be complex numbers with r 1 > 0, r 2 > z 1 = z 2 if n only if r 1 = r 2 n θ 1 = θ 2 + 2kπ for some integer k. 2. z 1 z 2 = r 1 r 2 e i(θ1+θ2). 3. z 1 z 2 = r1 r 2 e i(θ1 θ2). Proposition 7. If z = re iθ n k is positive integer, then z k = r k e ikθ. Proposition 8. Let z 1 n z 2 be complex numbers. Then rg(z 1 z 2 ) = rg(z 1 ) + rg(z 2 ). Proposition 9. (n Definition) Let z 0 be non-zero complex number with z 0 = r 0 e iθ0 n r 0 = z 0. Let n be positive integer. A complex number z stisfies z n = z 0 if n only if for some k = 0, 1,..., n 1 we hve z = n r 0 exp(i( θ 0 n + 2kπ n )). We cll these complex numbers z, the n-th roots of z 0. In the specil cse z 0 = 1, we cll these complex numbers z, the n-th roots of unity. Note tht the symbol n r 0 enotes the unique positive rel number which is n n-th roots of r 0. We let z ( 1 n ) 0 enote the set of n-th roots of z 0. Definition 10. (n remrk) We will sometimes refer to comlex numbers s points. Let z 0 be point. The ɛ neighborhoo of z 0 is the set of points given by z z 0 < ɛ. The elete ɛ neighborhoo of z 0 is the set of points given by 0 < z z 0 < ɛ.

3 Definition 11. Let S be subset of the set of complex numbers numbers. Let z 0 be point. We sy tht z 0 is n interior point of S if n only if there exists n ɛ neighborhoo of z 0 which is subset of S. We sy tht z 0 is n exterior point of S if n only if there exists n ɛ neighborhoo of z 0 which is subset of the complement of S. We sy tht z 0 is bounry point of S if n only if z 0 is neither n interior point is of S nor n exterior point of S. The set of bounry points of S is clle the bounry of S. We sy tht S is open if n only if no bounry point of S is n element of S. We sy tht S is close if n only if ech bounry point of S is n element of S. The closure of S is the union of S n the set of bounry points of S. Proposition 12. Let S be subset of the set of complex numbers numbers. S is open if n only if ech point of S is n interior point of S. S is close if n only if the closure of S is S. Definition 13. (n remrk) Let S be n open subset of the set of complex numbers numbers. We sy tht S is connecte if n only if ny two points of S cn be joine by finite union of line segments joine en to en tht lie entirely in S. This is not the usul topologicl efninion, but is equivlent in this setting. A nonempty, connecte, open set is clle omin. Definition 14. Let S be subset of the set of complex numbers numbers. We sy tht S is boune if n only if there is positive rel number B such tht for ll z S, we hve z < B. Definition 15. Let S be subset of the set of complex numbers numbers, n let w be complex number. We sy tht w is n ccumultion point of S if n only every elete neighborhoo of w contins t lest one point of S. Proposition 16. Let S be subset of the set of complex numbers numbers. S is close if n only if every ccumultion point of S is n element of S. Definition 17. (n remrk) We will let enote the set of complex numbers. We will stuy functions f : D where D is subset of. The set D is clle the omin of the function. We my think of function s rule which ssigns to ech element of the omin unique complex number. If the omin is not specifie, we ssume the omin is the set of ll complex numbers for which the rule mkes sense. Observe tht for ny f : D, there exists unique pir of rel vlue functions u, v of the two vribles x, y such tht for ll complex numbers in D. f(x + iy) = u(x, y) + iv(x, y) Definition 18. Let f : D where D is subset of. Suppose tht z 0 is complex number, n z 0 is n ccumultion point of D. Recll tht this hols if there is elete neighborhoo of z 0 which is subset of D. Let w 0 be complex number. We sy tht lim z z0 f(z) = w 0 if n only if for every ɛ > 0, there exists δ > 0 such tht for ll z D which re in the elete δ neighborhoo of z 0, we hve tht f(z) w 0 < ɛ. Proposition 19. Let f : D where D is subset of. Suppose tht z 0 = x 0 + iy 0 is complex number, n z 0 is n ccumultion point of D. Let w 0 = u 0 + iv 0 be complex number. Then lim z z0 f(z) = w 0 if n only if both lim (x,y) (x0,y 0) u(x, y) = u 0 n lim (x,y) (x0,y 0) v(x, y) = v 0. Remrk 20. We hve the sme theorems for limits of sums, proucts, n quotients tht hol for rel vlue functions of rel vrible. See Theorem 2 pge 48 of the text for precise sttement. Definition 21. Let f : D where D is subset of. Suppose tht z 0 is complex number n z 0 is n ccumultion point of D. We sy tht lim z z0 f(z) = if n only if for every ɛ > 0, there exists δ > 0 such tht for ll z D which re in the elete δ neighborhoo of z 0, we hve tht f(z) > 1 ɛ. Definition 22. Let f : D where D is subset of. Suppose tht there is positive number B such tht the set of z with z > B is subset of D. Let w 0 be complex number. We sy tht lim z f(z) = w 0 if n only if for every ɛ > 0, there exists δ > 0 such tht for ll z with z > 1 δ we hve tht f(z) w 0 < ɛ.

4 Definition 23. Let f : D where D is subset of. Suppose tht there is positive number B such tht the set of z with z > B is subset of D. We sy tht lim z f(z) = if n only if for every ɛ > 0, there exists δ > 0 such tht for ll z with z > 1 δ we hve tht f(z) > 1 ɛ. Proposition 24. If lim z z0 1 f(z) = 0, then lim z z 0 f(z) =. If lim z 0 f( 1 z ) = w 0, then lim z f(z) = w 0. If lim z 0 1 f( 1 z ) = 0, then lim z f(z) =. Definition 25. Let f : D where D is subset of. Let z 0 D. We sy tht f is continuous t z 0 if n only if for every ɛ > 0, there exists δ > 0 such tht for ll z D with z z 0 < δ we hve f(z) f(z 0 ) < ɛ. Proposition 26. Let f : D where D is subset of. Let z 0 D, n suppose tht z 0 is n ccumultion point of D. Then f is continuous t z 0 if n only if lim z z0 f(z) = f(z 0 ). Definition 27. Let f : D where D is subset of. Suppose tht z 0 is n interior point of D. We sy tht f is ifferentible t z 0 if n only if the limit f(z) f(z 0 ) lim z z 0 z z 0 exists n is some complex number w 0. In this cse w 0 is clle the erivtive of f t z 0, n we use the nottion f (z 0 ) = w 0. Proposition 28. Let f : D where D is subset of. Suppose tht z 0 is n interior point of D. Then f is ifferentible t z 0 if n only if the limit f(z 0 + h) f(z 0 ) lim h 0 h exists n is some complex number w 0. In this cse f (z 0 ) = w 0. Theorem 29. Let f : D where D is subset of. Suppose tht z 0 = x 0 + iy 0 is n interior point of D. Suppose tht f(x + iy) = u(x, y) + iv(x, y) for ll points of D. If f is ifferentible t z 0, then the first prtil erivtes of u n v exist t (x 0, y 0 ) n we hve u x (x 0, y 0 ) = v y (x 0, y 0 ), u y (x 0, y 0 ) = v x (x 0, y 0 ). Moreover, in this cse we hve f (z 0 ) = u x (x 0, y 0 ) + iv x (x 0, y 0 ). Definition 30. The equtions re clle the uchy-riemnn equtions. u x (x 0, y 0 ) = v y (x 0, y 0 ), u y (x 0, y 0 ) = v x (x 0, y 0 ) Theorem 31. Let f : D where D is subset of. Suppose tht z 0 = x 0 + iy 0 is n interior point of D. Suppose tht f(x + iy) = u(x, y) + iv(x, y) for ll points of D. Suppose tht the first prtil erivtes of u n v exist in n ɛ neighborhoo of (x 0, y 0 ) n re continuous t (x 0, y 0 ). Finlly, suppose tht the uchy-riemnn equtions u x (x 0, y 0 ) = v y (x 0, y 0 ), u y (x 0, y 0 ) = v x (x 0, y 0 ). re stisfie. Then f (z 0 ) exists, n f (z 0 ) = u x (x 0, y 0 ) + iv x (x 0, y 0 ). Theorem 32. (Polr form of uchy-riemnn equtions). Let f : D where D is subset of. Suppose tht z 0 = r 0 e iθ0 is n interior point of D. Suppose tht for z D with z = re iθ, we hve f(z) = u(r, θ) + iv(r θ). Suppose tht the first prtil erivtives of u n v with respect to r n θ exist in n ɛ neighborhoo of z 0 n re continuous t z 0. Finlly, suppose tht the equtions ru r (r 0, θ 0 ) = v θ (r 0, θ 0 ), u θ (r 0, θ 0 ) = rv r (r 0, θ 0 ). re stisfie. Then f (z 0 ) exists, n f (z 0 ) = e iθ0 (u r (r 0, θ 0 ) + iv r (r 0, θ 0 )).

5 Definition 33. Let f : D where D is subset of. If S is n open subset of D, we sy tht f is nlytic on S if n only if f (z) exists t ech z S. If z 0 is n interior point of D, we sy tht f is nlytic t z 0 if n only if f is nlytic on some neighborhoo of z 0. Finlly, if D = n f is nlytic on, we sy tht f is n entire function. Theorem 34. Let f : D where D is subset of. Suppose tht D is omin ( nonempty, open, connecte set). If f (z) = 0 for ll z D, then f is constnt on D. Definition 35. Let f : D where D is subset of. Suppose tht z 0 D. If every neighborhoo of z 0 inclues point z such tht f is nlytic t z, but f is not nlytic t z 0, we sy tht z 0 is singulr point of f. Definition 36. Let h : D R where D is subset of the xy plne. We sy tht h is hrmonic in D if n only if h hs continuous prtil erivtives of the first n secon orer n h xx + h yy = 0 everywhere in D. This eqution is known s Lplce s eqution. Theorem 37. Let f : D where D is subset of. Suppose tht D is omin ( nonempty, open, connecte set). Suppose tht f(x + iy) = u(x, y) + iv(x, y) for ll points of D. If f is nlytic on D, then u n v re hrmonic in D. orollry 38. Let f : D where D is subset of. Suppose tht D is omin ( nonempty, open, connecte set). Suppose tht f is nlytic on D, n lso the conjugte f : D is nlytic on D. Then f is constnt on D. orollry 39. Let f : D where D is subset of. Suppose tht D is omin ( nonempty, open, connecte set). Suppose tht f is nlytic on D, n f(z) is constnt on D. Then f is constnt on D. Definition 40. For ny complex number, z = x + iy we efine e z = e x e iy. We sometimes write exp(z) for e z. Proposition 41. The function e z is entire, n z ez = e z for ll complex numbers z. Proposition 42. Let z n w be complex numbers. We hve: 1. e z+w = e z e w. 2. e z+2πi = e z. 3. e z w = ez e w. 4. (e z ) n = e nz for ny positive integer n. Proposition 43. Let z = x + iy n w = ρe iθ 0. Then e z = w if n only if for some integer n. z = ln ρ + i(θ + 2nπ) Proposition 44. Let A enote the set of complex numbers x + iy with π < y π. Let B enote the set of non-zero complex numbers. Let h : A B be efine by h(z) = e z. Let g : B A be efine by g(w) = ln w + i Arg(w). Then h n g re inverse functions of ech other. Definition 45. For ny non-zero complex number w we efine Log(w) = ln w + i Arg(w) n log(w) = ln w + i rg(w). Also, we sy tht Log(w) is the principl vlue of log(w). Note tht log(w) is multiple-vlue function. Now, fix rel number α. We my obtin single vlue function from log(w) by tking the unique vlue of rg(w) with α < rg(w) < α + 2π. This single vlue function is clle brnch of log(w). Proposition 46. For ny non-zero complex number w, we hve log(w) = {z : e z = w}. Proposition 47. z Log(z) = 1 z for ll z 0 with π < Arg(z) < π. More generlly, if α is rel number n log(z) enote the corresponing brnch, then z log(z) = 1 z for ll z 0 with α < rg(z) < α + 2π.

6 Definition 48. For ny non-zero complex number z n ny complex number c we efine z c = exp(c log z). Also, the principl vlue of z c is efine to be exp(clogz). Proposition 49. For ny non-zero complex number z n ny positive integer n, the set z 1 n of n-th roots of z. is coincies with the set Proposition 50. Fix complex number c, n fix brnch of log(z). Let f(z) = z c enote the (single-vlue) function ttine by using this brnch in the efinition. Then f is nlytic on the omin given by z 0 n α < rg(z) < α + 2π. Moreover f (z) = cz c 1. Proposition 51. Fix complex number c 0, n fix one vlue of log(c). Let f(z) = c z enote the (single-vlue) function ttine. Then f is entire, n f (z) = c z log(c). Definition 52. The sine n cosine functions re efine by sin z = eiz e iz 2i Proposition 53. The sine n cosine functions re entire, n sin z = cos z, z, cos z = eiz + e iz. 2 cos z = sin z. z Remrk 54. The 4 other trigonometric functions of complex vrible cn be efine in terms of the sine n cosine functions. Moreover, the stnr trigonometric ientities for rel vrible continue to hol for complex vrible. Proposition 55. For ny complex number z = x + iy, sin z = sin x cosh(y) + i cos x sinh(y) cos z = cos x cosh(y) i sin x sinh(y). Definition 56. The hyperbolic sine n cosine functions re efine by sinh(z) = ez e z, cosh(z) = ez + e z. 2 2 Proposition 57. The hyperbolic sine n cosine functions re entire, n Proposition 58. Let z be complex number. sinh(z) = cosh(z), z Proposition 59. For ny complex number z = x + iy, Proposition 60. For ny complex number z, cosh(z) = sinh(z). z sinh(z) = i sin(iz), cosh(z) = cos(iz). sinh(z) = sinh(x) cos(y) + i cosh(x) sin(y) cosh(z) = cosh(x) cos(y) + i sinh(x) sin(y). sin(z) 2 = sin 2 (x) + sinh 2 (y) cos(z) 2 = cos 2 (x) + sinh 2 (y). Proposition 61. All zeros of the sine n cosine functions efine on the complex plne re rel. So sin(z) = 0 if n only if z = nπ for some integer n, n cos(z) = 0 if n only if z = π 2 + nπ for some integer n.

7 Definition 62. The multiple vlue inverse sine, cosine, n tngent functions re efine by sin 1 z = {w : sin w = z} cos 1 z = {w : cos w = z} tn 1 z = {w : tn w = z}. Proposition 63. We hve sin 1 z = i log(iz + (1 z 2 ) 1 2 ) cos 1 z = i log(z + i(1 z 2 ) 1 2 ) tn 1 z = i 2 log(i + z i z ). Definition 64. Let w be complex vlue function of rel vrible t. Let w(t) = u(t) + iv(t). If u (t) n v (t) exist we set w (t) = u (t) + iv (t). Proposition 65. Let z be complex vlue function of rel vrible t, sn suppose tht z (t) exists. Let f : D where D is subset of. Suppose tht f is nlytic t z(t). Let w be efine by w(t) = f(z(t)). Then w (t) = f (z(t)) z (t). Definition 66. Let w be complex vlue function of rel vrible t. Let w(t) = u(t) + iv(t). Suppose tht u n v re integrble on n intervl [, b]. We set w(t)t = u(t)t + i v(t)t. Proposition 67. Let w be complex vlue function of rel vrible t. Suppose tht W is complex vlue function of rel vrible t, such tht W (t) = w(t) for ll t in the intervl [, b]. Then w(t)t = W (b) W (). Definition 68. Let be subset of the complex plne. We sy tht is n rc if n only if there exists complex vlue function z, given by z(t) = x(t) + iy(t), efine on n intervl [, b] such tht x n y re continuous functions, n z if n only z = z(t) for some t [, b]. The function z is clle prmetric representtion of the rc. If the function z is one to one we sy tht is simple rc. If z(b) = z() but otherwise z is one to one we sy tht is simple close curve. If the functions x n y re continuously ifferentible on the intervl [, b], we sy tht is ifferentible rc. If is ifferentible rc such tht for ll t [, b] we hve z (t) 0, we sy tht is smooth rc. A contour is n rc consiting of finite number of smooth rcs joine en to en. If is simple close curve n contour, we sy tht is simple close contour. Definition 69. Let be smooth rc with prmetric representtion given by z(t) = x(t) + iy(t) for t [, b]. Let f be complex vlue function of complex vrible z. Suppose tht the function f(z(t)) z (t) is integrble on the intervl [, b]. We efine the contour integrl of f long by f(z)z = f(z(t)) z (t)t. If is contour, consiting of smooth rcs 1,..., n joine en to en, we efine the contour integrl by f(z)z = f(z)z f(z)z. n

8 Definition 70. Let be contour with prmetric representtion given by z(t) = x(t) + iy(t) for t [, b]. Let f be complex vlue function of complex vrible z. Suppose tht the function f(z(t)) is piecewise continuous on the intervl [, b]. This mens tht this function hs only finitely mny points of iscontinuity n the one-sie limits exist t these points. We sy in this sitution tht f is pieceswise continuous on. Theorem 71. If is contour n f is piecewise continuous on, then f(z)z is inepenent of the prmetric represention, provie the representtions re s given in Definition 69, n trce out the curve exctly once n in the sme irection. Theorem 72. Suppose tht is contour n f is piecewise continuous on. Let enote contour which trces the sme set of points in the opposite irection. Then f(z)z = f(z)z. Definition 73. Let be contour with prmetric representtion given by z(t) = x(t) + iy(t) for t [, b]. The length of is efine s L = z (t) t. Proposition 74. In the previous efinition, the length of is inepenent of the prmetric representtion. Proposition 75. Let w be piecewise continuous complex vlue function of rel vrible t on the intervl [, b]. Then w(t)t w(t) t. Theorem 76. Suppose tht is contour with length L, n f is piecewise continuous on. Suppose tht M is positive rel number such tht f(z) M for ll points z on where f(z) is efine. Then f(z)z ML. Moreover, with the hypothesis bove, such number M lwys exists. Theorem 77. Let f : D where D is subset of. Suppose tht D is omin ( nonempty, open, connecte set). Suppose tht f hs n ntierivtive F on D. (This mens tht F (z) = f(z) for ll z D.) Let A n B enote points of D. If is contour from A to B which lies entirely in D, then f(z)z = F (B) F (A). Theorem 78. Let f : D where D is subset of. Suppose tht D is omin ( nonempty, open, connecte set). Suppose tht f is continuous t ll points of D. The following re equivlent: () f hs n ntierivtive F on D. (b) If A n B re points of D n is contour from A to B which lies entirely in D, then f(z)z = F (B) F (A). (c) If is close contour lying entirely in D, then f(z)z = 0. Theorem 79. (uchy-gourst Theorem) Let f : D where D is subset of. If f is nlytic t ll points on n insie simple close contour, then f(z)z = 0. (Recll Definition 33.) Definition 80. Suppose tht D is omin ( nonempty, open, connecte set). We sy tht D is simply connecte if n only if for every simple close contour lying entirely in D, ll points insie re lso points of D. Theorem 81. Let f : D where D is simply connecte omin. Suppose tht f is nlytic t ll points of D. If is close contour lying entirely in D, then f(z)z = 0. orollry 82. Let f : D where D is subset of. Suppose tht 1 n 2 re simple close contours oriente in the sme irection, n one of these contours lies insie the other. Suppose tht f is nlytic t ll points on n between these two contours. Then 1 f(z)z = 2 f(z)z. Theorem 83. (uchy Integrl Formul) Let f : D where D is subset of. Suppose tht f is nlytic t ll points on n insie simple close contour, which is oriente in the positive (counter clockwise) irection. If z 0 is ny point insie, then f(z 0 ) = 1 2πi f(z) z z 0 z.

9 Theorem 84. (Extension of the uchy Integrl Formul) Let f : D where D is subset of. Suppose tht f is nlytic t ll points on n insie simple close contour, which is oriente in the positive irection. If n is non-negtive integer, n z 0 is ny point insie, then f (n) (z 0 ) exists, n f (n) (z 0 ) = n! f(z) z. 2πi (z z 0 ) n+1 orollry 85. Let f : D where D is subset of. If f is nlytic t some point z, then erivtives of ll orers exist n re nlytic t z. Theorem 86. (Liouville s theorem) Suppose tht f : is entire n boune. Then f is constnt function. Theorem 87. (Funmentl Theorem of Algebr) Suppose tht p : is polynomil of egree n 1. Then p(z) cn be expresse s prouct of liner fctors p(z) = c(z z 1 )(z z 2 ) (z z n ). Theorem 88. (Mximum Moulus Principle) Suppose tht D is omin ( nonempty, open, connecte set), n f : D is nlytic n not constnt. Then f(z) oes not ttin mximum vlue t ny point of D.