What's New
About the Project
NIST
4 Elementary FunctionsLogarithm, Exponential, Powers

§4.9 Continued Fractions

Contents

§4.9(i) Logarithms

4.9.1 ln(1+z)=z1+z2+z3+4z4+4z5+9z6+9z7+,
|ph(1+z)|<π.
4.9.2 ln(1+z1-z)=2z1-z23-4z25-9z27-16z29-,

valid when z\(-,-1][1,); see Figure 4.23.1(i).

For other continued fractions involving logarithms see Lorentzen and Waadeland (1992, pp. 566–568). See also Cuyt et al. (2008, pp. 196–200).

§4.9(ii) Exponentials

For z,

4.9.3 ez =11-z1+z2-z3+z2-z5+z2-
=1+z1-z2+z3-z2+z5-z2+z7-
=1+z1-(z/2)+z2/(43)1+z2/(415)1+z2/(435)1+z2/(4(4n2-1))1+
4.9.4 ez-en-1(z)=znn!-n!z(n+1)+z(n+2)-(n+1)z(n+3)+2z(n+4)-(n+2)z(n+5)+3z(n+6)-,

where

4.9.5 en(z)=k=0nzkk!.

For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563–564). See also Cuyt et al. (2008, pp. 193–195).

§4.9(iii) Powers

See Cuyt et al. (2008, pp. 217–220).