AngouriMath
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LogarithmOfAPower(AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity,AngouriMath.Core.ApproachFrom)
Method (no overloads)
Summary
A logarithm of a power to a constant real exponent, log(f^c), as c log(f), or as
c log(-f) + log((-1)^c) where f is negative near the point.
c log(-f) + log((-1)^c) where f is negative near the point.
Remarks
For f positive and c real, f^c is positive and ln(f^c) is c ln(f). For f negative, f^c is
(-f)^c (-1)^c on the principal branch, a positive number times one of modulus 1, so
ln(f^c) is c ln(-f) + ln((-1)^c), which is c ln(f) only for c in (-1, 1]. In that range the
first form holds wherever f is real, whatever its sign, so a two-sided limit whose sides
differ in sign can still be asked through it. Nothing read ln(f^c) through ln(f):
log(x, arcsin(x)^tan(1/2)) at 0, which is tan(1/2) from each side, was left unevaluated
after three seconds from the right and four two-sided, where log(x, arcsin(x)) was 1 at
once. Found by limitcheck atDEPTH=3 , SEED=65 ; no issue was filed.
(-f)^c (-1)^c on the principal branch, a positive number times one of modulus 1, so
ln(f^c) is c ln(-f) + ln((-1)^c), which is c ln(f) only for c in (-1, 1]. In that range the
first form holds wherever f is real, whatever its sign, so a two-sided limit whose sides
differ in sign can still be asked through it. Nothing read ln(f^c) through ln(f):
log(x, arcsin(x)^tan(1/2)) at 0, which is tan(1/2) from each side, was left unevaluated
after three seconds from the right and four two-sided, where log(x, arcsin(x)) was 1 at
once. Found by limitcheck at
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