Digital Library of Mathematical Functions
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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.4 Special Values and Limits

Contents

§4.4(i) Logarithms

4.4.1 ln1 =0,
4.4.2 ln(-1±0) =±π,
4.4.3 ln(±) =±12π.

§4.4(ii) Powers

4.4.4 0 =1,
4.4.5 ±π =-1,
4.4.6 ±π/2 =±,
4.4.7 2πk =1,
k,
4.4.8 ±π/3 =12±32,
4.4.9 ±2π/3 =-12±32,
4.4.10 ±π/4 =12±12,
4.4.11 ±3π/4 =-12±12,
4.4.12 ± =π/2.

§4.4(iii) Limits

4.4.13 limxx-alnx=0,
a>0,
4.4.14 limx0xalnx=0,
a>0,
4.4.15 limxxa-x=0,
4.4.16 limzza-z=0,
|phz|12π-δ (<12π),

where a () and δ ((0,12π]) are constants.

4.4.17 limn(1+zn)n=z,
z= constant.
4.4.18 limn(1+1n)n=.
4.4.19 limn((k=1n1k)-lnn)=γ=0.57721 56649 01532 86060,

where γ is Euler’s constant; see (5.2.3).