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-valuedCh.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