Digital Library of Mathematical Functions
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4 Elementary FunctionsTrigonometric Functions

§4.25 Continued Fractions

4.25.1 tanz=z1-z23-z25-z27-,
z±12π, ±32π, .
4.25.2 tan(az)=atanz1+(1-a2)tan2z3+(4-a2)tan2z5+(9-a2)tan2z7+,
|z|<12π, az±12π,±32π,.
4.25.3 arcsinz1-z2=z1-12z23-12z25-34z27-34z29-,

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

4.25.4 arctanz=z1+z23+4z25+9z27+16z29+,

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

4.25.5 2aarctan(1/z)=1+2az-a+a2+13z+a2+45z+a2+97z+,

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).