Chapter 3. Elementary Functions Weiqi Luo (骆伟祺) School of Software Sun Yat-Sen University Email:weiqi.luo@yahoo.com Office:# A313 Chapter 3: Elementary Functions The Exponential Functions The Logarithmic Function Branches and Derivatives of Logarithms Some Identities Involving Logarithms Complex Exponents Trigonometric Function Hyperbolic Functions Inverse Trigonometric and Hyperbolic Functions 2 School of Software 29. The Exponential Function The Exponential Function e e e , z x iy z x iy Single-Valued According to the Euler’ Formula e iy cos y i sin y u(x,y) v(x,y) e e cos y ie sin y z x x Note that here when x=1/n (n=2,3…) & y=0, e1/n denotes the positive nth root of e. 3 School of Software 29. The Exponential Function Properties z1 e e Let e z2 z1 z 2 z1 x1 iy1 ; z 2 x 2 +iy 2 e x 1 +iy1 e x 2 +iy 2 (e 1 e x iy1 Real value: x )( e 2 e ( e e )(e e x1 e x2 x1 x 2 e iy1 e iy 2 x1 ) e e x2 =e x1 x 2 Refer to pp. 18 iy 2 ) iy e 1e iy 2 e i(y 1 y 2 ) i(y 1 y 2 ) z1 z 2 ( x1 x 2 )+ i ( y1 y 2 ) z 1 +z 2 4 School of Software 29. The Exponential Function Properties e z1 z 2 e z2 e z1 e z1 e z2 e z1 z 2 e z2 Refer to Example 1 in Sec 22, (pp.68), we have that d e e z z everywhere in the z plane dz which means that the function ez is entire. 5 School of Software 0 29. The Exponential Function Properties e 0 z e e e z x iy re i For any complex number z r e & y x r | e | e 0 & arg( e ) y 2 n ( n 0, 1, 2, ...) z e z 2 i x e e z z 2 i e z 2 i e ,e z 2 i cos 2 i sin 2 1 which means that the function ez is periodic, with a pure imaginary period of 2πi 6 School of Software 29. The Exponential Function Properties e 0 x For any real value x while ez can be a negative value, for instance e i cos i sin 1 7 School of Software 29. The Exponential Function Example In order to find numbers z=x+iy such that e 1 i z e e e z x e x x iy 2&e 1 2e iy ln 2 & y i / 4 e i / 4 2 n , ( n 0, 1, 2, ...) 2 1 4 1 z ln 2 i ( 2 n ), ( n 0, 1, 2, ...) 2 4 8 School of Software 29. Homework pp. 92-93 Ex. 1, Ex. 6, Ex. 8 9 School of Software 30. The Logarithmic Function The Logarithmic Function log z ln r i ( 2 n ), ( n 0, 1, 2, ...) z re i 0 Please note that the Logarithmic Function is the multiple-valued function. ln r i z re ln r i ( 2 ) i One to infinite values ln r i ( 2 ) … It is easy to verify that e log z e ln r i ( 2 n ) e 10 ln r e i ( 2 n ) re i z School of Software 30. The Logarithmic Function The Logarithmic Function log z ln r i ( 2 n ), ( n 0, 1, 2, ...) z re i 0 ln | z | i arg( z ) Suppose that 𝝝 is the principal value of argz, i.e. -π <𝝝 ≤π L o g z ln r iA rg ( z ) ln r i is single valued. And log z L ogz i 2 n , n 0, 1, 2, ... 11 School of Software 30. The Logarithmic Function Example 1 log( 1 log( 1 3 i ) log(2 e 3i ) ? i ( 2 / 3 ) ln 2 i ( ) 2 2 n ), n 0, 1, 2... 3 12 School of Software 30. The Logarithmic Function Example 2 & 3 log 1 ln 1 i (0 2 n ) 2 n i , n 0, 1, 2, ... L og 1 0 log( 1) ln 1 i ( 2 n ) (2 n 1) i , n 0, 1, 2, ... L og ( 1) i 13 School of Software 31. Branches and Derivatives of Logarithms The Logarithm Function log z ln r i ( 2 n ), n 0, 1, 2, ... where𝝝=Argz, is multiple-valued. If we let θ is any one of the value in arg(z), and let α denote any real number and restrict the value of θ so that 2 The above function becomes single-valued. log z ln r i , ( r 0, 2 ) With components u ( r , ) ln r & v ( r , ) 14 School of Software 31. Branches and Derivatives of Logarithms The Logarithm Function log z ln r i , ( r 0, 2 ) is not only continuous but also analytic throughout the domain r 0, 2 A connected open set ? 15 School of Software 31. Branches and Derivatives of Logarithms The derivative of Logarithms log z ln r i , ( r 0, 2 ) u ( r , ) ln r & v ( r , ) ru r v & u rv r d log z e i dz d dz L og z ( u r iv r ) e i ( 1 r i 0) 1 re i 1 z 1 z 16 School of Software 31. Branches and Derivatives of Logarithms Examples When the principal branch is considered, then Log ( i ) Log ( i ) 3 ln 1 i 2 And 3 L og ( i ) 3(ln 1 i ) 2 i 2 3 i 2 Log ( i ) 3 Log ( i ) 3 17 School of Software 31. Homework pp. 97-98 Ex. 1, Ex. 3, Ex. 4, Ex. 9, Ex. 10 18 School of Software 32. Some Identities Involving Logarithms log( z1 z 2 ) log z1 log z 2 z1 r1 e where i 1 0 & z 2 r2 e i log( z1 z 2 ) log( r1 e 1 r2 e i 2 i 2 0 ) ln( r1 r2 ) i ( 1 2 2 n ) ln r1 ln r2 i ( 1 2 n1 ) i ( 2 2 n 2 ) [ln r1 i ( 1 2 n1 )] [ln r2 i ( 2 2 n 2 )] (ln | z1 | i arg z1 ) (ln | z 2 | i arg z 2 ) log z1 log z 2 log( z1 z2 n n1 n 2 1 1 ) log( z1 z 2 ) log z1 log z 2 log z1 log z 2 19 School of Software 32. Some Identities Involving Logarithms Example z1 z 2 1 log( z1 z 2 ) log(1) 2 n i log( z1 ) log( z 2 ) log( 1) (2 n 1) i log z1 log z 2 (2 n1 1) i (2 n 2 1) i 2( n1 n 2 1) i 2 n i log( z1 z 2 ) 20 n n1 n 2 1 School of Software 32. Some Identities Involving Logarithms When z≠0, then z e n n log z 1 z 1/ n e z e c ( n 0, 1, 2, ...) log z n c log z ( n 1, 2, 3 ...) Where c is any complex number 21 School of Software 32. Homework pp. 100 Ex. 1, Ex. 2, Ex. 3 22 School of Software 33. Complex Exponents Complex Exponents When z≠0 and the exponent c is any complex number, the function zc is defined by means of the equation z e c c log z where logz denotes the multiple-valued logarithmic function. Thus, zc is also multiple-valued. The principal value of zc is defined by z e c cL og z 23 School of Software 33. Complex Exponents If z re i and α is any real number, the branch log z ln r i ( r 0, 2 ) Of the logarithmic function is single-valued and analytic in the indicated domain. When the branch is used, it follows that the function z exp( c log z ) c is single-valued and analytic in the same domain. d dz z c d exp( c log z ) dz c exp( c log z ) z 24 School of Software 33. Complex Exponents Example 1 i log i ln 1 i ( 2 i exp( 2 i log i ) 2 n ) (2 n 2 i 2 i 1 ) i , ( n 0, 1, 2, ...) 2 exp[(4 n 1) ], ( n 0, 1, 2, ...) Note that i-2i are all real numbers 25 School of Software 33. Complex Exponents Example 2 The principal value of (-i)i is exp( iL og ( i )) exp( i (ln 1 i 2 P.V. i i exp )) exp 2 2 26 School of Software 33. Complex Exponents Example 3 The principal branch of z2/3 can be written exp( 2 L ogz ) exp( 3 2 ln r 3 2 i ) 3 3 2 r exp( i 2 ) 3 Thus 2 P.V. z3 3 2 r cos 2 i r sin 3 2 3 2 3 This function is analytic in the domain r>0, -π<𝝝<π 27 School of Software 33. Complex Exponents Example 4 Consider the nonzero complex numbers z1 1 i , z 2 1 i & z 3 1 i When principal values are considered ( z1 z 2 ) 2 e i i z3 e i / 4 e iL og (1 i ) e iL og ( 1 i ) e z2 e e iLog 2 iLog (1 i ) z1 e i i /4 e e 3 / 4 i ln 2 ( z 2 z 3 ) ( 2) e i iL og ( - 2) ( z1 z 2 ) z1 z 2 i (ln 2 ) / 2 i i (ln 2 ) / 2 e i i i ( z2 z3 ) z 2 z3 e i i e i 2 i (ln 2 ) / 2 28 School of Software e i ln 2 33. Complex Exponents The exponential function with base c c e z z log c Based on the definition, the function cz is multiple-valued. And the usual interpretation of ez (single-valued) occurs when the principal value of the logarithm is taken. The principal value of loge is unity. When logc is specified, cz is an entire function of z. d dz c z d e z log c e z log c log c c log c z dz 29 School of Software 33. Homework pp. 104 Ex. 2, Ex. 4, Ex. 8 30 School of Software 34. Trigonometric Functions Trigonometric Functions Based on the Euler’s Formula e ix cos x i sin x & e e e ix sin x ix ix & cos x iz sin z 2i cos x i sin x e e 2i e e ix ix Here x and y are real numbers 2 iz e e iz & cos z iz Here z is a complex number 2 31 School of Software 34. Trigonometric Functions Trigonometric Functions e e iz sin z iz e e iz & cos z 2i iz 2 Both sinz and cosz are entire since they are linear combinations of the entire Function eiz and e-iz d dz sin z cos z & d cos z sin z dz 32 School of Software 34. Homework pp.108-109 Ex. 2, Ex. 3 33 School of Software 35. Hyperbolic Functions Hyperbolic Function e e z sinh z z e e z , cosh z 2 z 2 Both sinhz and coshz are entire since they are linear combinations of the entire Function eiz and e-iz d dz d sinh z cosh z , cosh z sinh z dz 34 School of Software 35. Hyperbolic Functions Hyperbolic v.s. Trgonometric i sinh( iz ) sin z & cosh( iz ) cos z i sin( iz ) sinh z & cos( iz ) cosh z 35 School of Software 35. Homework pp. 111-112 Ex. 3 36 School of Software 36. Inverse Trigonometric and Hyperbolic Functions In order to define the inverse sin function sin-1z, we write 1 w sin sin w z e iw e z sin w z When iw ( e ) 2 iz ( e ) 1 0 iw 2i e iw 2 iz (1 z ) 2 1/ 2 w sin Similar, we get cos 1 tan 1 iw 1 z i log( iz (1 z ) 2 1/ 2 z i log( z i (1 z ) 2 1/ 2 z i 2 lo g i z iz ) Multiple-valued functions. One to infinite many values Note that when specific branches of the square root and logarithmic functions are used, all three Inverse functions become single-valued and analytic. 37 ) School of Software 36. Inverse Trigonometric and Hyperbolic Functions Inverse Hyperbolic Functions sinh cosh tanh 1 z log[ z ( z 1) 1 z log[ z ( z 1) 1 2 2 z 1 2 log 1/ 2 1/ 2 ] ] 1 z 1 z 38 School of Software 36. Homework pp. 114-115 Ex. 1 39 School of Software