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Showing content from https://dlmf.nist.gov/4.25 below:

§4.25 Continued Fractions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.25 Continued Fractions ⓘ
Keywords:
continued fractions, inverse trigonometric functions, trigonometric functions
Notes:
See Jones and Thron (1980, pp. 202–203), Wall (1948, pp. 343–349).
Permalink:
http://dlmf.nist.gov/4.25
See also:
Annotations for Ch.4
4.25.1 tan ⁡ z = z 1 − z2 3 − z2 5 − z2 7 − ⁢ ⋯ , z ≠ ± 1 2 ⁢ π , ± 3 2 ⁢ π , … . ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, tan ⁡ z : tangent function and z : complex variable
A&S Ref:
4.3.94
Permalink:
http://dlmf.nist.gov/4.25.E1
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Annotations for §4.25 and Ch.4
4.25.2 tan ⁡ ( a ⁢ z ) = a ⁢ tan ⁡ z 1 + ( 1 − a2 ) ⁢ tan2 ⁡ z 3 + ( 4 − a2 ) ⁢ tan2 ⁡ z 5 + ( 9 − a2 ) ⁢ tan2 ⁡ z 7 + ⁢ ⋯ , | ℜ ⁡ z | < 1 2 ⁢ π , a ⁢ z ≠ ± 1 2 ⁢ π , ± 3 2 ⁢ π , … . ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, ℜ ⁡ : real part, tan ⁡ z : tangent function, a : real or complex constant and z : complex variable
A&S Ref:
4.3.95
Permalink:
http://dlmf.nist.gov/4.25.E2
Encodings:
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See also:
Annotations for §4.25 and Ch.4
4.25.3 arcsin ⁡ z 1 − z2 = z 1 − 1 ⋅ 2 ⁢ z2 3 − 1 ⋅ 2 ⁢ z2 5 − 3 ⋅ 4 ⁢ z2 7 − 3 ⋅ 4 ⁢ z2 9 − ⁢ ⋯ , ⓘ
Symbols:
arcsin ⁡ z : arcsine function and z : complex variable
A&S Ref:
4.4.44
Permalink:
http://dlmf.nist.gov/4.25.E3
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Annotations for §4.25 and Ch.4

valid when z lies in the open cut plane shown in Figure 4.23.1(i).

4.25.4 arctan ⁡ z = z 1 + z2 3 + 4 ⁢ z2 5 + 9 ⁢ z2 7 + 16 ⁢ z2 9 + ⁢ ⋯ , ⓘ
Symbols:
arctan ⁡ z : arctangent function and z : complex variable
A&S Ref:
4.4.43
Referenced by:
§3.10(ii)
Permalink:
http://dlmf.nist.gov/4.25.E4
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Annotations for §4.25 and Ch.4

valid when z lies in the open cut plane shown in Figure 4.23.1(ii).

4.25.5 e 2 ⁢ a ⁢ arctan ⁡ ( 1 / z ) = 1 + 2 ⁢ a z − a + a2 + 1 3 ⁢ z + a2 + 4 5 ⁢ z + a2 + 9 7 ⁢ z + ⁢ ⋯ , ⓘ
Symbols:
e : base of natural logarithm, arctan ⁡ z : arctangent function, a : real or complex constant and z : complex variable
A&S Ref:
4.2.42
Permalink:
http://dlmf.nist.gov/4.25.E5
Encodings:
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See also:
Annotations for §4.25 and Ch.4

valid when z lies in the open cut plane shown in Figure 4.23.1(iv).

See Lorentzen and Waadeland (1992, pp. 560–571) for other continued fractions involving inverse trigonometric functions. See also Cuyt et al. (2008, pp. 201–203, 205–210).

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