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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.9 Continued Fractions

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Referenced by:
§3.10(ii)
Permalink:
http://dlmf.nist.gov/4.9
See also:
Annotations for Ch.4
Contents
  1. §4.9(i) Logarithms
  2. §4.9(ii) Exponentials
  3. §4.9(iii) Powers

§4.9(i) Logarithms

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Keywords:
continued fractions, logarithm function
Notes:
See Wall (1948, pp. 342–343).
Permalink:
http://dlmf.nist.gov/4.9.i
See also:
Annotations for §4.9 and Ch.4
4.9.1 ln⁡(1+z)=z1+z2+z3+4⁢z4+4⁢z5+9⁢z6+9⁢z7+⁢⋯,
|ph⁡(1+z)|<π.
4.9.2 ln⁡(1+z1−z)=2⁢z1−z23−4⁢z25−9⁢z27−16⁢z29−⁢⋯,
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Symbols:
ln⁡z: principal branch of logarithm function and z: complex variable
A&S Ref:
4.1.40
Permalink:
http://dlmf.nist.gov/4.9.E2
Encodings:
TeX, pMML, png
See also:
Annotations for §4.9(i), §4.9 and Ch.4

valid when z∈ℂ∖(−∞,−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

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Keywords:
continued fractions, exponential function
Notes:
See Wall (1948, p. 348).
Permalink:
http://dlmf.nist.gov/4.9.ii
See also:
Annotations for §4.9 and Ch.4

For z∈ℂ,

4.9.3 ez =11−z1+z2−z3+z2−z5+z2−⁢⋯
=1+z1−z2+z3−z2+z5−z2+z7−⁢⋯
=1+z1−(z/2)+z2/(4⋅3)1+z2/(4⋅15)1+z2/(4⋅35)1+⁢⋯⁢z2/(4⁢(4⁢n2−1))1+⁢⋯
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Symbols:
e: base of natural logarithm, n: integer and z: complex variable
A&S Ref:
4.2.40
Permalink:
http://dlmf.nist.gov/4.9.E3
Encodings:
TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:
Annotations for §4.9(ii), §4.9 and Ch.4
4.9.4 ez−en−1⁡(z)=znn!−n!⁢z(n+1)+z(n+2)−(n+1)⁢z(n+3)+2⁢z(n+4)−⁢(n+2)⁢z(n+5)+3⁢z(n+6)−⁢⋯,

where

4.9.5 en⁡(z)=∑k=0nzkk!.
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Defines:
en⁡(z): expansion of exponential (locally)
Symbols:
!: factorial (as in n!), k: integer, n: integer and z: complex variable
Permalink:
http://dlmf.nist.gov/4.9.E5
Encodings:
TeX, pMML, png
See also:
Annotations for §4.9(ii), §4.9 and Ch.4

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

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Permalink:
http://dlmf.nist.gov/4.9.iii
See also:
Annotations for §4.9 and Ch.4

See Cuyt et al. (2008, pp. 217–220).

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