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Showing content from https://dlmf.nist.gov/4.6 below:

§4.6 Power Series ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions

§4.6 Power Series ⓘ
Permalink:
http://dlmf.nist.gov/4.6
See also:
Annotations for Ch.4
Contents
  1. §4.6(i) Logarithms
  2. §4.6(ii) Powers
§4.6(i) Logarithms ⓘ
Keywords:
logarithm function, power series
Notes:
See Hardy (1952, pp. 471–473) for (4.6.1). The other equations are variations of this.
Permalink:
http://dlmf.nist.gov/4.6.i
See also:
Annotations for §4.6 and Ch.4
4.6.1 ln ⁡ ( 1 + z ) = z − 1 2 ⁢ z2 + 1 3 ⁢ z3 − ⋯ , | z | ≤ 1 , z ≠ − 1 , ⓘ
Symbols:
ln ⁡ z : principal branch of logarithm function and z : complex variable
A&S Ref:
4.1.24
Referenced by:
§4.45(i), §4.6(i), (8.15.2)
Permalink:
http://dlmf.nist.gov/4.6.E1
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(i), §4.6 and Ch.4
4.6.2 ln ⁡ z = ( z − 1 z ) + 1 2 ⁢ ( z − 1 z ) 2 + 1 3 ⁢ ( z − 1 z ) 3 + ⋯ , ℜ ⁡ z ≥ 1 2 , ⓘ
Symbols:
ln ⁡ z : principal branch of logarithm function, ℜ ⁡ : real part and z : complex variable
A&S Ref:
4.1.25
Permalink:
http://dlmf.nist.gov/4.6.E2
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(i), §4.6 and Ch.4
4.6.3 ln ⁡ z = ( z − 1 ) − 1 2 ⁢ ( z − 1 ) 2 + 1 3 ⁢ ( z − 1 ) 3 − ⋯ , | z − 1 | ≤ 1 , z ≠ 0 , ⓘ
Symbols:
ln ⁡ z : principal branch of logarithm function and z : complex variable
A&S Ref:
4.1.26
Permalink:
http://dlmf.nist.gov/4.6.E3
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(i), §4.6 and Ch.4
4.6.4 ln ⁡ z = 2 ⁢ ( ( z − 1 z + 1 ) + 1 3 ⁢ ( z − 1 z + 1 ) 3 + 1 5 ⁢ ( z − 1 z + 1 ) 5 + ⋯ ) , ℜ ⁡ z ≥ 0 , z ≠ 0 , ⓘ
Symbols:
ln ⁡ z : principal branch of logarithm function, ℜ ⁡ : real part and z : complex variable
A&S Ref:
4.1.27
Permalink:
http://dlmf.nist.gov/4.6.E4
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(i), §4.6 and Ch.4
4.6.5 ln ⁡ ( z + 1 z − 1 ) = 2 ⁢ ( 1 z + 1 3 ⁢ z3 + 1 5 ⁢ z5 + ⋯ ) , | z | ≥ 1 , z ≠ ± 1 , ⓘ
Symbols:
ln ⁡ z : principal branch of logarithm function and z : complex variable
A&S Ref:
4.1.28
Permalink:
http://dlmf.nist.gov/4.6.E5
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(i), §4.6 and Ch.4
4.6.6 ln ⁡ ( z + a ) = ln ⁡ a + 2 ⁢ ( ( z 2 ⁢ a + z ) + 1 3 ⁢ ( z 2 ⁢ a + z ) 3 + 1 5 ⁢ ( z 2 ⁢ a + z ) 5 + ⋯ ) , a > 0 , ℜ ⁡ z ≥ − a , z ≠ − a . ⓘ
Symbols:
ln ⁡ z : principal branch of logarithm function, ℜ ⁡ : real part, a : real or complex constant and z : complex variable
A&S Ref:
4.1.29
Permalink:
http://dlmf.nist.gov/4.6.E6
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(i), §4.6 and Ch.4
§4.6(ii) Powers ⓘ
Notes:
See Hardy (1952, pp. 476–477).
Referenced by:
Erratum (V1.0.11) for Clarifications
Permalink:
http://dlmf.nist.gov/4.6.ii
Addition (effective with 1.0.11):
A sentence was added after (4.6.7) to explain that it is a generalization of (1.2.2) using (1.2.6)
See also:
Annotations for §4.6 and Ch.4
Binomial Expansion ⓘ
Keywords:
binomial expansion
See also:
Annotations for §4.6(ii), §4.6 and Ch.4
4.6.7 ( 1 + z ) a = 1 + a 1 ! ⁢ z + a ⁢ ( a − 1 ) 2 ! ⁢ z2 + a ⁢ ( a − 1 ) ⁢ ( a − 2 ) 3 ! ⁢ z3 + ⋯ , ⓘ
Symbols:
! : factorial (as in n ! ), a : real or complex constant and z : complex variable
Referenced by:
§4.6(ii), §4.6(ii)
Permalink:
http://dlmf.nist.gov/4.6.E7
Encodings:
TeX, pMML, png
See also:
Annotations for §4.6(ii), §4.6(ii), §4.6 and Ch.4

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).

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