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 -1Ch.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