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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.6 Power Series

  1. §4.6(i) Logarithms
  2. §4.6(ii) Powers

§4.6(i) Logarithms

4.6.1 ln(1+z)=z12z2+13z3,
|z|1, z1,
4.6.2 lnz=(z1z)+12(z1z)2+13(z1z)3+,
4.6.3 lnz=(z1)12(z1)2+13(z1)3,
|z1|1, z0,
4.6.4 lnz=2((z1z+1)+13(z1z+1)3+15(z1z+1)5+),
z0, z0,
4.6.5 ln(z+1z1)=2(1z+13z3+15z5+),
|z|1, z±1,
4.6.6 ln(z+a)=lna+2((z2a+z)+13(z2a+z)3+15(z2a+z)5+),
a>0, za, za.

§4.6(ii) Powers

Binomial Expansion

4.6.7 (1+z)a=1+a1!z+a(a1)2!z2+a(a1)(a2)3!z3+,

valid when a is any real or complex constant and |z|<1. If a=0,1,2,, then the series terminates and z is unrestricted. Note that (4.6.7) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6).