Ch.3:Elementary Functions Ch.3:Elementary Functions LOutline 3.1 Polynomials and Rational Functions Chapter 3:Elementary Functions 3.2 The Exponential,Trigonometric,and Hyperbolic Functions 3.2.1 The Complex Function e* 3.2.2 Trigonometric Functions Li,Yongzhao 3.2.3 Hyperbolic Functions State Key Laboratory of Integrated Services Networks,Xidian University 3.3 The Logarithmic Function September 28,2010 3.5 Complex Powers and Inverse Trigonometric Functions Ch.3:Elementary Functions Ch.3:Elementary Functions 3.1 Polynomials and Rationol Functions 3.1 Polynomials and Rational Functions The Degree of the Polynomial and Rational Functions Deflation of Polynomial Functions The degree of the polynomial which has the form of You can always divide a"dividend"polynomial by a "divisor" pn(z)=a0+a12+a222+..+an2n polynomial to obtain a "quotient"polynomial and a "reminder"polynomial whose degree is less than that of the is n if the complex constant an is nonzero divisor The rational function which has the form of dividend=divisor x quotient +remainder R0-器 If z1 is any arbitrary complex number,then division of pn(z) by the degree-one polynomial z-z1 must result in a has numerator degree m and denominator degree n,if am0 remainder of lower degree:in other words,a constant, and bn≠0 We will begin our study with these two simple types of Pn(z)=(z-z1)pn-1(z)+constant (1) functions where the quotient polynomial pn-1(z)has degree n-1
Ch.3: Elementary Functions Chapter 3: Elementary Functions Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University September 28, 2010 Ch.3: Elementary Functions Outline 3.1 Polynomials and Rational Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.1 The Complex Function ez 3.2.2 Trigonometric Functions 3.2.3 Hyperbolic Functions 3.3 The Logarithmic Function 3.5 Complex Powers and Inverse Trigonometric Functions Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions The Degree of the Polynomial and Rational Functions The degree of the polynomial which has the form of pn(z) = a0 + a1z + a2z2 + ... + anzn is n if the complex constant an is nonzero The rational function which has the form of Rm,n(z) = a0 + a1z + a2z2 + ... + amzm b0 + b1z + b2z2 + ... + bnzn has numerator degree m and denominator degree n, if am = 0 and bn = 0 We will begin our study with these two simple types of functions Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Deflation of Polynomial Functions You can always divide a ”dividend” polynomial by a ”divisor” polynomial to obtain a ”quotient” polynomial and a ”reminder” polynomial whose degree is less than that of the divisor dividend = divisor × quotient + remainder If z1 is any arbitrary complex number, then division of pn(z) by the degree-one polynomial z − z1 must result in a remainder of lower degree: in other words, a constant, pn(z)=(z − z1)pn−1(z) + constant (1) where the quotient polynomial pn−1(z) has degree n − 1
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.1 Polynomials and Ration Functions L3.1 Polynomials and Rational Functions Deflation of Polynomial Functions(Cont'd) Zeros of Polynomial Functions If z1 happens to be a zero of pn(z),we deduce that the In order to deflate a polynomial function,we must find the remainder is zero.Thus(1)shows how z-z1 has been zeros first.Hence the two questions arise:1)How to find a factored out from pn(z).We say pn(z)has been"deflated" zero of pn(z);2)How do we know pn(z)has any zeros? If z2 is a zero of the quotient pn-1(z),we can deflate further Gauss helped us answer the second question in his doctoral by factoring out z-z2.and so on,until we run out of zeros, dissertation of 1799:Every nonconstant polynomial with leaving us with the factorization complex coefficients has at least one zero in C p(z)=(z-)(z-22)…(z-2)pm-k(z)(2) We immediately conclude that a polynomial of degree n has n zeros,since we can continue to factor out zeros in the Example on page 99-100 gives us an explicit explanation of deflation process until we reach the final,constant,quotient. how this procedure works Repeated zeros are counted according to their multiplicities Ch.3:Elementary Functions Ch.3:Elementary Functions 3.1 Polynomials and Rationol Functions 3.1 Polynomials and Rational Functions Zeros-Characterization of Polynomials Taylor Form of the Polynomials With the issue of existence of zeros for the quotients settled Any polynomial function pn(z)can be expressed in the form we have a complete factorization of any polynomial as follows of Taylor form centered at zo as follows pm(2）=am(z-21)(z-22)…(z-2n)） This equation demonstrates that a polynomial of degree n has -ω+e-++m-or n zeros and pn(z)is completed determined by its zeros,up to Pn(z） .01 1！ (k) a constant multiple fan =ae-0 The Fundamental Theorem only tell us there are zeros,it k=0 doesn't tell us how to find them We use the nomenclature Maclaurin Form for the Taylor form The cases of degree one the two are simple,but the higher degree is very difficult or unsolvable centered at zo =0
Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Deflation of Polynomial Functions (Cont’d) If z1 happens to be a zero of pn(z), we deduce that the remainder is zero. Thus (1) shows how z − z1 has been factored out from pn(z). We say pn(z) has been ”deflated” If z2 is a zero of the quotient pn−1(z), we can deflate further by factoring out z − z2, and so on, until we run out of zeros, leaving us with the factorization pn(z)=(z − z1)(z − z2)···(z − zk)pn−k(z) (2) Example on page 99-100 gives us an explicit explanation of how this procedure works Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Zeros of Polynomial Functions In order to deflate a polynomial function, we must find the zeros first. Hence the two questions arise: 1) How to find a zero of pn(z); 2) How do we know pn(z) has any zeros? Gauss helped us answer the second question in his doctoral dissertation of 1799: Every nonconstant polynomial with complex coefficients has at least one zero in C We immediately conclude that a polynomial of degree n has n zeros, since we can continue to factor out zeros in the deflation process until we reach the final, constant, quotient. Repeated zeros are counted according to their multiplicities Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Zeros-Characterization of Polynomials With the issue of existence of zeros for the quotients settled we have a complete factorization of any polynomial as follows pn(z) = an(z − z1)(z − z2)···(z − zn) This equation demonstrates that a polynomial of degree n has n zeros and pn(z) is completed determined by its zeros, up to a constant multiple {an} The Fundamental Theorem only tell us there are zeros, it doesn’t tell us how to find them The cases of degree one the two are simple, but the higher degree is very difficult or unsolvable Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Taylor Form of the Polynomials Any polynomial function pn(z) can be expressed in the form of Taylor form centered at z0 as follows pn(z) = pn(z0) 0! + pn(z0) 1! (z − z0)1 + ··· + p(n) n (z0) n! (z − z0)n = n k=0 p(k) n (z0) k! (z − z0)k We use the nomenclature Maclaurin Form for the Taylor form centered at z0 = 0
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.1 Polynomials and Ration Functions L3.1 Polynomials and Rational Funsctions Factored Form of Rational Functions Factored Form of Rational Functions(Cont'd) Since the rational functions are ratios of polynomials,all the previous conclusions can be applied to their numerators and The zeros of the numerator are,of course,zeros of Rm.n(z); denominators separately zeros of the denominator are called poles of Rm.n(z) Probably the most enlightening display comes from the Zeros and poles can,of course,be multiple factored from Rmne)=m2-2-2)-e-2m Clearly,the magnitude of Rm.n(z)grows without bounds as z approaches a poles bn(z-E1)(z-2)…(z-En) where designates the zeros of the numerator and {E} With the knowledge of poles,we can express Rm.n(z)in terms of partial fractions which will be discussed subsequently designates those of the denominator(We assume the common zeros have been canceled) 白·0+之。，急，是2风C 4口0。+t+生+意0c Ch.3:Elementary Functions Ch.3:Elementary Functions 3.1 Polynomials and Rationol Functions 3.1 Polynomials and Rational Functions Partial Fractional Decomposition How to Find the Coefficients(A} a0十a1z+a2z2+..+dn2m IfR is a rational function The brute-force procedure consists in rearranging the whose denominator degree n=di+d2+...+d,exceeds its proposed form(3)over a common denominator and numerator degree m,then Rm.n has a partial fraction comparing the resulting numerator,term be term,with the decomposition of the from original numerator of Rm.n.But this will result in solving a 4 1) group of linear equations A quicker,more sophisticated method for evaluating the -1 +-a西+.…+司 (3) (A is illustrated in the example on page 106 十…十 F+…+-器 The deduced conclusion is if Rm.n can be written in the form (3).then a general expression for the coefficients is 1 ds[ where the A are constants(The 's are assumed -)Rmn(a创 A)=lim distinct) 口
Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Factored Form of Rational Functions Since the rational functions are ratios of polynomials, all the previous conclusions can be applied to their numerators and denominators separately Probably the most enlightening display comes from the factored from Rm,n(z) = am(z − z1)(z − z2)···(z − zm) bn(z − ξ1)(z − ξ2)···(z − ξn) where {zk} designates the zeros of the numerator and {ξk} designates those of the denominator (We assume the common zeros have been canceled) Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Factored Form of Rational Functions (Cont’d) The zeros of the numerator are, of course, zeros of Rm,n(z); zeros of the denominator are called poles of Rm,n(z) Zeros and poles can, of course, be multiple Clearly, the magnitude of Rm,n(z) grows without bounds as z approaches a poles With the knowledge of poles, we can express Rm,n(z) in terms of partial fractions which will be discussed subsequently Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions Partial Fractional Decomposition If Rm,n = a0+a1z+a2z2+...+amzm bn(z−ξ1)d1 (z−ξ2)d2 ···(z−ξr)dr is a rational function whose denominator degree n = d1 + d2 + ··· + dr exceeds its numerator degree m, then Rm,n has a partial fraction decomposition of the from Rm,n = A(1) 0 (z−ξ1)d1 + A(1) 1 (z−ξ1)d1−1 + ··· + A(1) d1−1 (z−ξ1) + A(2) 0 (z−ξ2)d2 + ··· + A(1) d2−1 (z−ξ2) + ··· + A(r) 0 (z−ξr)dr + ··· + A(r) dr−1 (z−ξr) (3) where the {A(j) s } are constants (The ξk’s are assumed distinct) Ch.3: Elementary Functions 3.1 Polynomials and Rational Functions How to Find the Coefficients {A(j) s } The brute-force procedure consists in rearranging the proposed form (3) over a common denominator and comparing the resulting numerator, term be term, with the original numerator of Rm,n. But this will result in solving a group of linear equations A quicker, more sophisticated method for evaluating the {A(j) s } is illustrated in the example on page 106 The deduced conclusion is if Rm,n can be written in the form (3), then a general expression for the coefficients is A(j) s = lim z→ξj 1 s! ds dzs (z − ξj )djRm,n(z)
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.2 The Exponential Trigonometric,and Hyperboic Functions L3.2 The Exponential,Trigonometric,and Hyperboc Functions L3.2.1 The Complex Function e= L3.2.1 The Complex Function e The Complex Function e2 Polar Form of the Complex Function e The complex exponential function es plays a prominent role in The polar components of e*is analytic function theory,not only because of its own important properties but because it is used to define the |e1=e,arge=y+2kx(k=0,±1，±2，.) complex trigonometric and hyperbolic functions From the above expression,we can see that e=is never zero. If z=x+iy,es e(cosy+isiny)according to the Euler's However,e=does assume every other complex value Equation The exponential function is one-to-one on the real axis,but it e is an entire function and its arbitrary degree of derivative is is not one-to-one on the complex plane.In fact,we have itself,i.e., 1.The equation e==1 holds if,and only if,2=2kmi,where k is an integer e=e2=→ -e2=e 2.The equation e==e*holds if,and only if,21=22+2mi, where k is an integer Ch.3:Elementary Functions Ch.3:Elementary Functions 3.2 The Exponential,Trigonometric,and Hyperbolic Functions 3.2 The Exponential,Trigonometric,and Hyperbolic Functions L3.2.1 The Complex Function e= L3.2.2 Trigonometric Functions Polar Form of the Complex Function e*(Cont'd) Trigonometric Functions For real variables,we have the identities The Theorem tell us e=is periodic with complex period 2xi eiy-e-iy eiy+e-iy If we divide up the z-plane into the infinite horizontal strips: siny=2i cosy= 2 S={x+i-o<x<o,(2n-1)m<y≤(2n+1)x} We extend the identities to the complex case:Given any where n=0,±l,±2，.. complex number z,we define eiz-e-iz imz=2—, eiste-is e*behaves in the same manner on each strip.Furthermore,e C0s2:= 2 is one-to-one on each strip Since eis and e-is are entire functions,so are sin z and cos z. Any one of these strips is called a fundamental region for es Some further identities remain valid in the complex case(See page 113)
Ch.3: Elementary Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.1 The Complex Function ez The Complex Function ez The complex exponential function ez plays a prominent role in analytic function theory, not only because of its own important properties but because it is used to define the complex trigonometric and hyperbolic functions If z = x + iy, ez = ex(cos y + isin y) according to the Euler’s Equation ez is an entire function and its arbitrary degree of derivative is itself, i.e., d dz ez = ez =⇒ dn dzn ez = ez Ch.3: Elementary Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.1 The Complex Function ez Polar Form of the Complex Function ez The polar components of ez is |ez| = ex, arg ez = y + 2kπ (k = 0, ±1, ±2,...) From the above expression, we can see that ez is never zero. However, ez does assume every other complex value The exponential function is one-to-one on the real axis, but it is not one-to-one on the complex plane. In fact, we have 1. The equation ez = 1 holds if, and only if, z = 2kπi, where k is an integer 2. The equation ez1 = ez2 holds if, and only if, z1 = z2 + 2kπi, where k is an integer Ch.3: Elementary Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.1 The Complex Function ez Polar Form of the Complex Function ez (Cont’d) The Theorem tell us ez is periodic with complex period 2πi If we divide up the z-plane into the infinite horizontal strips: Sn := {x + iy|−∞ <x< ∞,(2n − 1)π<y ≤ (2n + 1)π} where n = 0, ±1, ±2,... ez behaves in the same manner on each strip. Furthermore, ez is one-to-one on each strip Any one of these strips is called a fundamental region for ez Ch.3: Elementary Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.2 Trigonometric Functions Trigonometric Functions For real variables, we have the identities sin y = eiy − e−iy 2i , cos y = eiy + e−iy 2 We extend the identities to the complex case: Given any complex number z, we define sin z := eiz − e−iz 2i , cos z := eiz + e−iz 2 Since eiz and e−iz are entire functions, so are sin z and cos z. Some further identities remain valid in the complex case (See page 113)
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.2 The Exponential,Trigonometric,and Hyporbolic Functions L3.2 The Exponential,Trigonometric,and Hyperbolic Functions L3.2.2 Trigonometric Functions L3.2.3 Hyperbolic Functions The Distinction Between the Real and Complex Cases Hyperbolic Functions The real cosine function is bounded by 1,i.e.. For any complex number z,we define cos<1,for all real r es-e-: e2+e-2 But in the complex case,the cosine function sinhz=2—,c0shz=2 1cos(训=拦| =coshy One nice feature of the complex variable perspective is that it reveals the intimate connection between hyperbolic functions which is unbounded and,in fact,is never less than 1 and their trigonometric analogues(See page 114-115 for But note that,this does not mean sin z or cos z is always details) greater than 1! +口·0+t。年之，220C Ch.3:Elementary Functions Ch.3:Elementary Functions 3.3 The Logarithmic Function 3.3 The Logarithmic Function Definition of Logarithmic Functions Definition of Logarithmic Functions(Cont'd) log z is defined as the inverse of the exponential function;i.e., Thus w=log z is also a multiple-valued function.The w=logz if z=et explicit definition is as follows Since et is never zero,we presume that z0.Let us write z Definition 3:If 20,then we define logz to be the set of in the polar form as z=rei and w in the standard form as infinitely many values w=u+iv.Then the equation z=et becomes re诏=eu+iw=e“ew logz:=Loglz+iargz =L0glz+iArgz+2km(k=0,土1，±2，.) Taking magnitudes of both sides we deduce that r =e",or that u is the ordinary logarithm of r:u Log r=Log z The multiple-valuedness of log z simply reflects the fact that The equality of the remaining factors,ei=ei,identifies v as the imaginary part of logarithm is the polar angle the (multiple-valued)polar angle 0=arg z:v=arg z =0 (multiple-valued);the real part is single-valued
Ch.3: Elementary Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.2 Trigonometric Functions The Distinction Between the Real and Complex Cases The real cosine function is bounded by 1, i.e., | cos x| ≤ 1, for all real x But in the complex case, the cosine function | cos(iy)| = e−y+ey 2 = cosh y which is unbounded and, in fact, is never less than 1 But note that, this does not mean |sin z| or | cos z| is always greater than 1! Ch.3: Elementary Functions 3.2 The Exponential, Trigonometric, and Hyperbolic Functions 3.2.3 Hyperbolic Functions Hyperbolic Functions For any complex number z, we define sinh z := ez − e−z 2 , cosh z := ez + e−z 2 One nice feature of the complex variable perspective is that it reveals the intimate connection between hyperbolic functions and their trigonometric analogues (See page 114-115 for details) Ch.3: Elementary Functions 3.3 The Logarithmic Function Definition of Logarithmic Functions log z is defined as the inverse of the exponential function; i.e., w = log z if z = ew Since ew is never zero, we presume that z = 0. Let us write z in the polar form as z = reiθ and w in the standard form as w = u + iv. Then the equation z = ew becomes reiθ = eu+iv = eueiv Taking magnitudes of both sides we deduce that r = eu, or that u is the ordinary logarithm of r: u = Log r = Log |z| The equality of the remaining factors, eiθ = eiv, identifies v as the (multiple-valued) polar angle θ=arg z: v = arg z = θ Ch.3: Elementary Functions 3.3 The Logarithmic Function Definition of Logarithmic Functions (Cont’d) Thus w = log z is also a multiple-valued function. The explicit definition is as follows Definition 3: If z = 0, then we define log z to be the set of infinitely many values log z : = Log|z| + i arg z = Log|z| + iArgz + i2kπ (k = 0, ±1, ±2,...) The multiple-valuedness of log z simply reflects the fact that the imaginary part of logarithm is the polar angle θ (multiple-valued); the real part is single-valued
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.3 The Logarithmic Function L3.3 The Logarithmic Function Properties of Logarithmic Functions The Principle Value of Logarithm Logz The notation of branch cut is used to resolve the ambiguity in the designation of the polar angle 0=arg z Many familiar properties of the real logarithmic function can We take Argz to be the principal value of arg z,in the interval be extended to the complex case,but it should be noted that (T,T+2 which shifts the 2-discontinuities to the ray 0=T log z is multiple-valued.Hence,the precise statements of Similarly,we generate single-valued branches of log z.The these extensions are more complicated principle value of the logarithm Logz is the value inherited lfz≠0，we have z=elog,but from the principal value of the argument: loge2=z+2kri(k=0,±1，±2，.) Logz :Loglz|+iArgz (Note that we use the same convention 'capital L'for the principal value as for the real value,since Argz =0 if z is positive real) Ch.3:Elementary Functions Ch.3:Elementary Functions 3.3 The Logarithmic Function 3.3 The Logarithmic Function Analyticity and Derivative of Logz Other Branches of logz Other branches log z can be employed if the location of the Logz also inherits,from Argz,the discontinuities along the discontinuities on the negative axis is inconvenient.Clearly. branch cut the specification However,at all points off the nonpositive real axis,Logz is Cr(z):=Logz +iargt z continuous when it is defined on the interval(-,and we have the following theorem results in a single-valued function whose imaginary part lies in Theorem 4:The function Logz is analytic in the domain D* the interval (T,+2 consisting of all points of the complex plane except those Also,Theorem 4 shows that this function is analytic in the lying on the nonpositive real axis.Furthermore complex plane excluding the ray and the origin d. When complex arithmetic is incorporated into computer 0g=2 for z in D* packages,all functions must of necessity be programmed as single-valued
Ch.3: Elementary Functions 3.3 The Logarithmic Function Properties of Logarithmic Functions Many familiar properties of the real logarithmic function can be extended to the complex case, but it should be noted that log z is multiple-valued. Hence, the precise statements of these extensions are more complicated If z = 0, we have z = elog z, but log ez = z + 2kπi (k = 0, ±1, ±2,...) Ch.3: Elementary Functions 3.3 The Logarithmic Function The Principle Value of Logarithm Logz The notation of branch cut is used to resolve the ambiguity in the designation of the polar angle θ = arg z We take Argz to be the principal value of arg z, in the interval (τ,τ + 2π] which shifts the 2π-discontinuities to the ray θ = τ Similarly, we generate single-valued branches of log z. The principle value of the logarithm Logz is the value inherited from the principal value of the argument: Logz := Log|z| + iArgz (Note that we use the same convention ’capital L’ for the principal value as for the real value, since Argz = 0 if z is positive real) Ch.3: Elementary Functions 3.3 The Logarithmic Function Analyticity and Derivative of Logz Logz also inherits, from Argz, the discontinuities along the branch cut However, at all points off the nonpositive real axis, Logz is continuous when it is defined on the interval (−π, π] and we have the following theorem Theorem 4: The function Logz is analytic in the domain D∗ consisting of all points of the complex plane except those lying on the nonpositive real axis. Furthermore d dzLogz = 1z , for z in D∗ Ch.3: Elementary Functions 3.3 The Logarithmic Function Other Branches of log z Other branches log z can be employed if the location of the discontinuities on the negative axis is inconvenient. Clearly, the specification Lτ (z) := Logz + i argτ z results in a single-valued function whose imaginary part lies in the interval (τ,τ + 2π] Also, Theorem 4 shows that this function is analytic in the complex plane excluding the ray θ = τ and the origin When complex arithmetic is incorporated into computer packages, all functions must of necessity be programmed as single-valued
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.5 Complex Powers and Ieverse Trigonometric Functions L3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions Complex Powers Functions(Cont'd) Since log z Logz +iArgz +2kmi,we can get the following One important theoretical use of the logarithmic function is to expression define complex powers of z.The definition is motivated by the identity =ea(Logl-l+iArgs+2ki)=e(Loglsl+iArgs)e2ki (4) 2”=(elog)”=enlog where=0,±l,±2，.(See Example1 on page132) which holds for any integer n The values of z obtained by taking k =k and =2 Definition 5:If a is a complex constant and 0,then we (in Eq.(4)will therefore be the same when define za by ea2kiri ea2kati 20 :=ealog By Theorem 3 of Sec.3.2 this occurs only if This means that each value of log z leads to a particular value a2kini a2k2xi+2mxi of z where m is an integer.By solving this equation,we get a=m/(k-2) Ch.3:Elementary Functions Ch.3:Elementary Functions 3.5 Complex Powers and Trigonometric Functions 3.5 Complex Powers and Inverse Trigon Complex Powers Functions(Cont'd) Complex Powers Functions(Cont'd) This means only when a is a real rational number,(4)yields some identical values of z Eq.(5)is entirely consistent with the theory of roots discussed If a is not a real rational number,we obtain infinitely many in Sec.1.5 different values for z,one for each choice of the integer k in In summary, Eq.(4) 2 is single-valued when o is a real integer One the other hand,if a =m/n,where m and n>0 are z takes finitely many values when a is a real rational number integers having no common factor,then one can verify that z takes infinitely many values in all other cases there are exactly n distinct values of zm/m,namely From Definition 4 and 5,we know that each branch of log z yields a branch of z.For example,using the principal branch 2=ep(Logl判ep(agz+2k✉) (5) of log z we obtain the principal branch of z,namely,eLogs where =0,1,...,n -1
Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions One important theoretical use of the logarithmic function is to define complex powers of z. The definition is motivated by the identity zn = elog zn = en log z which holds for any integer n Definition 5: If α is a complex constant and z = 0, then we define zα by zα := eα log z This means that each value of log z leads to a particular value of zα Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions (Cont’d) Since log z = Logz + iArgz + 2kπi, we can get the following expression zα = eα(Log|z|+iArgz+2kπi) = eα(Log|z|+iArgz)eα2kπi (4) where k = 0, ±1, ±2, ··· (See Example 1 on page 132) The values of zα obtained by taking k = k1 and k = k2 (= k1) in Eq.(4) will therefore be the same when eα2k1πi = eα2k2πi By Theorem 3 of Sec. 3.2 this occurs only if α2k1πi = α2k2πi + 2mπi where m is an integer. By solving this equation, we get α = m/(k1 − k2) Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions (Cont’d) This means only when α is a real rational number, (4) yields some identical values of zα If α is not a real rational number, we obtain infinitely many different values for zα, one for each choice of the integer k in Eq.(4) One the other hand, if α = m/n, where m and n > 0 are integers having no common factor, then one can verify that there are exactly n distinct values of zm/n, namely zm/n = exp mn Log|z| exp imn (Argz + 2kπ) (5) where k = 0, 1,...,n − 1 Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions (Cont’d) Eq.(5) is entirely consistent with the theory of roots discussed in Sec. 1.5 In summary, zα is single-valued when α is a real integer zα takes finitely many values when α is a real rational number zα takes infinitely many values in all other cases From Definition 4 and 5, we know that each branch of log z yields a branch of zα. For example, using the principal branch of log z we obtain the principal branch of zα, namely, eαLogz
Ch.3:Elementary Functions Ch.3:Elementary Functions L3.5 Complex Powers and leverse Trigonometric Functions L3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions(Cont'd) Inverse Trigonometric Functions We have exponentials expressed in terms of trig functions,trig Since e in entire and Logz is analytic in the slit domain D*, functions expressed as exponentials,and logs interpreted as the chain rule implies that the principal branch of z is also analytic in D* inverse of exponentials For z in D*,we have Similarly,we can get the inverse trigonometric functions for ()(aLogs) d complex numbers dz We start with the inverse sin function w=sin-2.From the Other branches of z can be constructed by using other identity:sinwwe can deduce that branches of log z,and since each branch of the latter has e2iw-2izeiw-1=0 derivative 1/z,the formula By solving the above quadratic formula,we arrive at 9=a e=iz+(1-z2)1/2 is valid for each corresponding branch of za Next,by taking logarithms,we get: sin-1 z=-ilogliz+(1-22)] Ch.3:Elementary Functions L3.5 Complex Powers and Imerse Trigonometric Functions Inverse Trigonometric Functions(Cont'd) We can obtain a branch of the multiple-valued function sin-lz by first choosing a branch of the square root and then selecting a suitable branch of the logarithm Using the chain rule and the formula of sin-1z,one can show that any such branch of sin-lz satisfies d 1 m)=0-m≠出 where the choice of the square root on the right must be the same as that used in the branch of sin-z The same methods can be applied to inverse cosine,tangent, and hyperbolic functions
Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Complex Powers Functions (Cont’d) Since ez in entire and Logz is analytic in the slit domain D∗, the chain rule implies that the principal branch of zα is also analytic in D∗ For z in D∗, we have d dz eαLogz = eαLogz ddz (αLogz) = eαLogz αz Other branches of zα can be constructed by using other branches of log z, and since each branch of the latter has derivative 1/z, the formula d dz (zα) = αzα 1z is valid for each corresponding branch of zα Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Inverse Trigonometric Functions We have exponentials expressed in terms of trig functions, trig functions expressed as exponentials, and logs interpreted as inverse of exponentials Similarly, we can get the inverse trigonometric functions for complex numbers We start with the inverse sin function w = sin−1 z. From the identity: z = sin w = eiw−e−iw 2i , we can deduce that e2iw − 2izeiw − 1=0 By solving the above quadratic formula, we arrive at eiw = iz + (1 − z2)1/2 Next, by taking logarithms, we get: sin−1 z = −ilog[iz + (1 − z2)1/2] Ch.3: Elementary Functions 3.5 Complex Powers and Inverse Trigonometric Functions Inverse Trigonometric Functions (Cont’d) We can obtain a branch of the multiple-valued function sin−1 z by first choosing a branch of the square root and then selecting a suitable branch of the logarithm Using the chain rule and the formula of sin−1 z, one can show that any such branch of sin−1 z satisfies d dz (sin−1 z) = 1 (1 − z2)1/2 (z = ±1) where the choice of the square root on the right must be the same as that used in the branch of sin−1 z The same methods can be applied to inverse cosine, tangent, and hyperbolic functions