MAT334H1-F – LEC0101 Complex Variables USUAL COMPLEX FUNCTIONS – 2 September 25th, 2020 Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 1 / 16 Complex power functions – 1 Similarly to the real case, for 푤 ∈ ℂ and 푧 ∈ ℂ ⧵ {0} we want to set 푧푤 ≔ 푒푤 log 푧. But then it is only defined up to a factor 푒2푖푛휋푤, 푛 ∈ ℤ, since log is well defined modulo 2푖휋. 푤 푤 log 푧 푤 Log 푧 2푖푛휋푤 푧 ≔ 푒 = {푒 푒 ∶ 푛 ∈ ℤ} Example 1 1 1 Log 푧 푖푛휋 Log 푧 For instance √푧 = 푧 2 = {푒 2 푒 ∶ 푛 ∈ ℤ} = {±푒 2 }. Indeed, the square root is well-defined only up to a sign. However there is no indeterminacy when 푤 ∈ ℤ since 푒2푖휋푛푤 = 1 when 푛푤 ∈ ℤ. Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 2 / 16 The power functions – 2 Beware 푤 푤 푤 2푖휋푤푛 The identity (푧1푧2) = 푧1 푧2 is only true modulo a factor 푒 , 푛 ∈ ℤ (i.e. as sets/multivalued functions). And…As if that wasn’t enough… BEWARE The identity 푧푤1+푤2 = 푧푤1 푧푤2 is generally false even as multivalued functions: • 푧푤1+푤2 is well-defined up to a factor 푒2푖휋푛(푤1+푤2), 푛 ∈ ℤ. • 푧푤1 푧푤2 is well-defined up to a factor 푒2푖휋(푛푤1+푘푤2), 푛, 푘 ∈ ℤ. So 푧푤1+푤2 ⊂ 푧푤1 푧푤2 (as multivalued functions). Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 3 / 16 The complex cosine and sine – 1 Definitions We define cos ∶ ℂ → ℂ and sin ∶ ℂ → ℂ respectively by 푒푖푧 + 푒−푖푧 푒푖푧 − 푒−푖푧 cos(푧) ≔ and sin(푧) ≔ 2 2푖 Proposition • ∀푧 ∈ ℂ, cos2 푧 + sin2 푧 = 1 • ∀푧, 푤 ∈ ℂ, sin(푧 + 푤) = sin 푧 cos 푤 + cos 푧 sin 푤 • ∀푧, 푤 ∈ ℂ, cos(푧 + 푤) = cos 푧 cos 푤 − sin 푧 sin 푤 • ∀푧 ∈ ℂ, sin(−푧) = − sin(푧) • ∀푧 ∈ ℂ, cos(−푧) = cos(푧) • ∀푧 ∈ ℂ, sin(푧 + 2휋) = sin(푧) • ∀푧 ∈ ℂ, cos(푧 + 2휋) = cos(푧) Homework: prove some of them. Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 4 / 16 The complex cosine and sine – 2 Proposition The functions cos ∶ ℂ → ℂ and sin ∶ ℂ → ℂ are surjective. Proof. Let 푤 ∈ ℂ, we look for 푧 ∈ ℂ such that cos(푧) = 푤, or equivalently 푒푖푧 + 푒−푖푧 = 2푤. Set 푢 = 푒푖푧 then the above equation becomes 푢 + 푢−1 = 2푤 or equivalently 푢2 − 2푤푢 + 1 = 0. Take such a 푢 (which is non-zero) then, since the range of exp is ℂ ⧵ {0}, there exists 푧 ∈ ℂ such that 푢 = 푒푖푧. ■ Homework What is wrong with this proof? 푖푧 −푖푧 푖푧 −푖푧 푒푖푧 푒−푖푧 Let 푧 ∈ ℂ, then |cos(푧)| = 푒 +푒 ≤ 푒 + 푒 = | | + | | = 1 + 1 = 1. | 2 | | 2 | | 2 | 2 2 2 2 Hence ∀푧 ∈ ℂ, | cos 푧| ≤ 1. This property is obviously false according to the above proposition. Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 5 / 16 The complex cosine and sine – 3 푐 −푐 −푐 푐 The horizontal line ℑ(푧) = 푐 is mapped by cos to cos(푥) 푒 +푒 + 푖 sin(푥) 푒 −푒 ∶ 푥 ∈ ℝ { ( 2 ) ( 2 ) } which is an ellipse (possibly flat for 푐 = 0). ℑ ℑ cos ℜ ℜ Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 6 / 16 The complex cosine and sine – 4 푦 −푦 −푦 푦 The vertical line ℜ(푧) = 푐 is mapped by cos to cos(푐) 푒 +푒 + 푖 sin(푐) 푒 −푒 ∶ 푦 ∈ ℝ { ( 2 ) ( 2 ) } which is a branch of hyperbola. ℑ ℑ cos ℜ ℜ Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 7 / 16 The complex cosine and sine – 5 ℑ ℑ cos ℜ ℜ Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 8 / 16 The complex cosine and sine – 6 Proposition 휋 • cos 푧 = 0 ⇔ ∃푛 ∈ ℤ, 푧 = + 휋푛 2 • sin 푧 = 0 ⇔ ∃푛 ∈ ℤ, 푧 = 휋푛 Proof. 푖푧 −푖푧 cos 푧 = 0 ⇔ 푒 + 푒 = 0 sin 푧 = 0 ⇔ 푒푖푧 − 푒−푖푧 = 0 2푖푧 ⇔ 푒 = −1 ⇔ 푒2푖푧 = 1 ⇔ ∃푛 ∈ ℤ, 2푖푧 = 푖휋 + 2푖휋푛 ⇔ ∃푛 ∈ ℤ, 2푖푧 = 2푖휋푛 휋 ⇔ ∃푛 ∈ ℤ, 푧 = + 휋푛 ⇔ ∃푛 ∈ ℤ, 푧 = 휋푛 2 ■ Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 9 / 16 Other complex trigonometric functions Definition: the complex tangent function We define tan ∶ ℂ ⧵ 휋 + 휋푛 ∶ 푛 ∈ ℤ → ℂ by { 2 } sin 푧 tan 푧 ≔ cos 푧 Definition: the complex cotangent function We define cot ∶ ℂ ⧵ {휋푛 ∶ 푛 ∈ ℤ} → ℂ by cos 푧 cot 푧 ≔ sin 푧 Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 10 / 16 Inverse trigonometric functions – 1 We want to find the inverse of cos. (cos is not injective so we will get a multivalued function as for log). 푖푤 2 푖푤 cos(푤) = 푧 ⇔ (푒 ) − 2푧푒 + 1 = 0 Then 푣2 − 2푧푣 + 1 = 0 ⇔ 푣 = 푧 + √푧2 − 1 Here √⋅ is already multivalued: it is only well defined up to its sign. Hence 푖푤 √ 2 √ 2 푒 = 푧 + 푧 − 1 ⇔ 푖푤 = log (푧 + 푧 − 1) √ 2 ⇔ 푤 = −푖 log (푧 + 푧 − 1) We may repeat the same thing for arcsin and arctan. Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 11 / 16 Inverse trigonometric functions – 2 √ 2 arccos(푧) = −푖 log (푧 + 푧 − 1) √ 2 arcsin(푧) = −푖 log (푖푧 + 1 − 푧 ) 푖 1 − 푖푧 arctan(푧) = log , 푧 ≠ ±푖 2 (1 + 푖푧) Beware They are multivalued functions defined on ℂ. Beware By the way, the range of tan is ℂ ⧵ {±푖}. Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 12 / 16 Inverse trigonometric functions – 3 We define the principal branches of arccos, arcsin and arctan by1: ℂ ⧵ ((−∞, −1] ∪ [1, +∞)) → {푧 ∈ ℂ ∶ 0 < ℜ(푧) < 휋} Arccos ∶ 2 푧 ↦ −푖 Log (푧 + √푧 − 1) 휋 휋 ℂ ⧵ ((−∞, −1] ∪ [1, +∞)) → {푧 ∈ ℂ ∶ − < ℜ(푧) < } Arcsin ∶ 2 2 2 푧 ↦ −푖 Log (푖푧 + √1 − 푧 ) 휋 휋 ℂ ⧵ ((−푖∞, −푖] ∪ [푖, +푖∞)) → {푧 ∈ ℂ ∶ − < ℜ(푧) < } Arctan ∶ 2 2 푧 ↦ 푖 Log 1−푖푧 2 ( 1+푖푧 ) 1We take the square root whose real part is non-negative. Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 13 / 16 Inverse trigonometric functions – 4 ℑ ℑ cos ℜ ℜ ℑ ℑ 0 휋 −1 1 Arccos 푖 tan ℑ ℑ ℜ ℜ − 휋 휋 2 2 Arctan −푖 sin ℜ ℜ − 휋 휋 −1 1 2 2 Arcsin Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 14 / 16 Inverse trigonometric functions – 5 arccos(푧) = {± Arccos(푧) + 2휋푛 ∶ 푛 ∈ ℤ} 푛 arcsin(푧) = {(−1) Arcsin(푧) + 휋푛 ∶ 푛 ∈ ℤ} arctan(푧) = {Arctan(푧) + 휋푛 ∶ 푛 ∈ ℤ} Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 15 / 16 The complex hyperbolic functions Definitions We define cosh ∶ ℂ → ℂ and sinh ∶ ℂ → ℂ respectively by 푒푧 + 푒−푧 푒푧 − 푒−푧 cosh(푧) ≔ and sinh(푧) ≔ 2 2 Proposition • ∀푧 ∈ ℂ, cos(푧) = cosh(푖푧) • ∀푧 ∈ ℂ, sin(푧) = −푖 sinh(푖푧) The above proposition allows us to derive hyperbolic identities from the trigonometric ones. Homework • ∀푧 ∈ ℂ, cosh2(푧) − sinh2(푧) = 1 • ∀푧, 푤 ∈ ℂ, cosh(푧 + 푤) = cosh 푧 cosh 푤 + sinh 푧 sinh 푤 • ∀푧, 푤 ∈ ℂ, sinh(푧 + 푤) = sinh 푧 cosh 푤 + cosh 푧 sinh 푤 • ∀푥, 푦 ∈ ℝ, cos(푥 + 푖푦) = cos 푥 cosh 푦 − 푖 sin 푥 sinh 푦 • ∀푥, 푦 ∈ ℝ, sin(푥 + 푖푦) = sin 푥 cosh 푦 + 푖 cos 푥 sinh 푦 Jean-Baptiste Campesato MAT334H1-F – LEC0101 – Sep 25, 2020 16 / 16.