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Elementary functions

# Elementary functions - following the Observe Soln u of v...

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Unformatted text preview: : following the Observe : Soln u. of v conjugate harmonic a find , ) , ( If Q. 2 2 y x x y x u + = u. of H.C. a is that v Conclude . v f(z) Im (iii) 0)}. {(0,- C D domain a in analytic is ) ( ) ( ). ( Re then , 1 ) ( If ) ( 2 2 y x y z f ii z f u z z f i +- = = = = = Chapter 3: Elementary Functions 1. Exponential Functions 2. Trigonometric Functions 3. Hyperbolic Functions 4. Logarithmic Functions 5. Complex Exponents 1. Inverse Trigonometric Functions 2. Inverse Hyperbolic Functions z n 3 2 e of series laurin' .... ! z ...... ! 3 z ! z z 1 2 + + + + + + = n 2 1 2 1 2 1 , z z z z z z e e e e- + = = , sin , cos iv u y) sin i (cosy e e . e e f(z) Let ) 2 ( x iy x iy x z y e v y e u e x x = = ⇒ + ≡ + = = = = + x y y x x y x x x y x x v u v u y e v y e v y e u y e u- = = ⇒ = =- = = ⇒ , cos , sin sin , cos continuous are , , , u clearly and satisfied are equations CR Thus x y x y v v u ( 29 z z z iy x x x x e e dz d e e e y e i y e v i is z f = ⇒ = = + = + = ′ ⇒ . sin cos u (z) f and able differenti ) ( x z. number complex any for e as 1 sin cos sin cos . e (3) x 2 2 z ≠ ⇒ ∈ 2200 = = ∴ = + = ⇒ + = = z x x z iy iy iy x e R x e e e y y e y i y e e e ( 29 . .......... 2 , 1 , , 2 e arg & , . e may write We z z ± ± = + = ∴ = = = = = n n y y e e when e e e z x i iy x π φ ρ ρ φ 1 2 sin 2 cos e Hence 2 sin & 1 2 cos ) 4 ( i 2 = + = = = π π π π π i x 1 sin cos- = + = π π π i e i ( 29 ( 29 1 sin cos- =- +- =- π π π i e i i i e i = + = 2 / sin 2 / cos 2 / π π π ( 29 ( 29 i i e i- =- +- =- 2 / sin 2 / cos 2 / π π π ... ,......... 3 , 2 , 1 , . 2 . . 5 2 2 2 = 2200 = ∴ ⇒ = = ± + n e e i period imaginary with periodic is e e e e e z i n z z z i z i z π π π π C z if negative be may x e x ∈ ℜ ∈ 2200 z e But ) 6 ( 1 e such that z Find : z- = Example π π π π π n y x n n y e e e e e x i iy x z 2 & ... 2 ,. 1 , , 2 and , 1 . 1 . 1 : Solution + = = ⇒ ± ± = + = = ⇒ = ⇒- = 1 e then 2,... 1, 0, n i, 1) (2n iy x z if Thus, z- = ± ± = + = + = π anywhere. analytic not is ) 7 ( : Excercise z e i + =- 1 e such that z of values all Find Q. 1 2z i iy x z e e e i e Solution 4 2 1 2 1 2 2 . 1 : π = ⇒ + =-- ,.. 2 , 1 , ; 2 4 2 , 2 1 2 ± ± = + = = ⇒- n n y e x π π ( 29 ( 29 ,.... 2 , 1 , , 8 2 ln 1 2 1 8 , 2 ln 1 2 1 ± ± = + + + = + = ∴ + = + = ⇒ n n i iy x z n y x π π π π . 2 sin , 2 e x cos then , real is x If (1) Functions ric Trigonomet ix i e e x e ix ix ix--- = + = , sin cos ) 1 ( 2 sin , 2 cos define we complex, is z If z i z e i e e z e e z iz iz iz iz iz + = ⇒---- = + =-- z z ec z z z z z z z z sin 1 cos , cos 1 sec , sin cos cot , cos sin tan = = = = 2. Since e z is analytic z and linear combination of two analytic functions is again analytic, hence it follows that sin z and cos z are analytic functions....
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Elementary functions - following the Observe Soln u of v...

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