About the Project
4 Elementary FunctionsLogarithm, Exponential, Powers

§4.6 Power Series


§4.6(i) Logarithms

4.6.1 ln(1+z)=z-12z2+13z3-,
|z|1, z-1,
4.6.2 lnz=(z-1z)+12(z-1z)2+13(z-1z)3+,
4.6.3 lnz=(z-1)-12(z-1)2+13(z-1)3-,
|z-1|1, z0,
4.6.4 lnz=2((z-1z+1)+13(z-1z+1)3+15(z-1z+1)5+),
z0, z0,
4.6.5 ln(z+1z-1)=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, z-a, z-a.

§4.6(ii) Powers

Binomial Expansion

4.6.7 (1+z)a=1+a1!z+a(a-1)2!z2+a(a-1)(a-2)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).