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The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface. A single valued version called the principal value of the logarithm can be defined which is discontinuous on the negative x axis and equals the multivalued version on a single branch cut.
The convention will be used here that a capital first letter is used for the principal value of functions and the lower case version refers to the multivalued function. The single valued version of definitions and identities is always given first followed by a separate section for the multiple valued versions.
ln(r) is the standard natural logarithm of the real number r.Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (-π, π].Arg(z) is the principal value of the arg function, its value is restricted to (-π, π]. It can be computed using Arg(x+iy)= atan2(y, x). Log ( z ) = ln ( | z | ) + i Arg ( z ) {\displaystyle \operatorname {Log} (z)=\ln(|z|)+i\operatorname {Arg} (z)} e Log ( z ) = z {\displaystyle e^{\operatorname {Log} (z)}=z}The multiple valued version of log(z) is a set but it is easier to write it without braces and using it in formulas follows obvious rules.
log(z) is the set of complex numbers v which satisfy ev = zarg(z) is the set of possible values of the arg function applied to z.When k is any integer:
log ( z ) = ln ( | z | ) + i arg ( z ) {\displaystyle \log(z)=\ln(|z|)+i\arg(z)} log ( z ) = Log ( z ) + 2 π i k {\displaystyle \log(z)=\operatorname {Log} (z)+2\pi ik} e log ( z ) = z {\displaystyle e^{\log(z)}=z}Principal value forms:
Ln ( 1 ) = 0 {\displaystyle \operatorname {Ln} (1)=0} Ln ( e ) = 1 {\displaystyle \operatorname {Ln} (e)=1}Multiple value forms, for any k an integer:
log ( 1 ) = 0 + 2 π i k {\displaystyle \log(1)=0+2\pi ik} log ( e ) = 1 + 2 π i k {\displaystyle \log(e)=1+2\pi ik}Principal value forms:
Log ( z 1 ) + Log ( z 2 ) = Log ( z 1 z 2 ) ( mod 2 π i ) {\displaystyle \operatorname {Log} (z_{1})+\operatorname {Log} (z_{2})=\operatorname {Log} (z_{1}z_{2}){\pmod {2\pi i}}} Log ( z 1 ) − Log ( z 2 ) = Log ( z 1 / z 2 ) ( mod 2 π i ) {\displaystyle \operatorname {Log} (z_{1})-\operatorname {Log} (z_{2})=\operatorname {Log} (z_{1}/z_{2}){\pmod {2\pi i}}}Multiple value forms:
log ( z 1 ) + log ( z 2 ) = log ( z 1 z 2 ) {\displaystyle \log(z_{1})+\log(z_{2})=\log(z_{1}z_{2})} log ( z 1 ) − log ( z 2 ) = log ( z 1 / z 2 ) {\displaystyle \log(z_{1})-\log(z_{2})=\log(z_{1}/z_{2})}A complex power of a complex number can have many possible values.
Principal value form:
z 1 z 2 = e z 2 Log ( z 1 ) {\displaystyle {z_{1}}^{z_{2}}=e^{z_{2}\operatorname {Log} (z_{1})}} Log ( z 1 z 2 ) = z 2 Log ( z 1 ) ( mod 2 π i ) {\displaystyle \operatorname {Log} {\left({z_{1}}^{z_{2}}\right)}=z_{2}\operatorname {Log} (z_{1}){\pmod {2\pi i}}}Multiple value forms:
z 1 z 2 = e z 2 log ( z 1 ) {\displaystyle {z_{1}}^{z_{2}}=e^{z_{2}\log(z_{1})}}Where k1, k2 are any integers:
log ( z 1 z 2 ) = z 2 log ( z 1 ) + 2 π i k 2 {\displaystyle \log {\left({z_{1}}^{z_{2}}\right)}=z_{2}\log(z_{1})+2\pi ik_{2}} log ( z 1 z 2 ) = z 2 Log ( z 1 ) + z 2 2 π i k 1 + 2 π i k 2 {\displaystyle \log {\left({z_{1}}^{z_{2}}\right)}=z_{2}\operatorname {Log} (z_{1})+z_{2}2\pi ik_{1}+2\pi ik_{2}}เมนูนำทาง
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