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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±i0) =±πi,
4.4.3 ln(±i) =±12πi.

§4.4(ii) Powers

4.4.4 e0 =1,
4.4.5 e±πi =-1,
4.4.6 e±πi/2 =±i,
4.4.7 e2πki =1,
k,
4.4.8 e±πi/3 =12±i32,
4.4.9 e±2πi/3 =-12±i32,
4.4.10 e±πi/4 =12±i12,
4.4.11 e±3πi/4 =-12±i12,
4.4.12 i±i =eπ/2.

§4.4(iii) Limits

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

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

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

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