Showing content from https://dlmf.nist.gov/4.33 below:
§4.33 Maclaurin Series and Laurent Series ⣠Hyperbolic Functions ⣠Chapter 4 Elementary Functions
§4.33 Maclaurin Series and Laurent Series â
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Keywords:
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Laurent series, Maclaurin series, hyperbolic functions
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Referenced by:
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§24.2(i)
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Permalink:
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http://dlmf.nist.gov/4.33
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See also:
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Annotations for Ch.4
4.33.1 sinh â¡ z = z + z3 3 ! + z5 5 ! + ⯠, â
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Symbols:
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! : factorial (as in n ! ), sinh â¡ z : hyperbolic sine function and z : complex variable
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A&S Ref:
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4.5.62
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Permalink:
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http://dlmf.nist.gov/4.33.E1
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Encodings:
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TeX, pMML, png
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Annotations for §4.33 and Ch.4
4.33.2 cosh â¡ z = 1 + z2 2 ! + z4 4 ! + ⯠. â
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Symbols:
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! : factorial (as in n ! ), cosh â¡ z : hyperbolic cosine function and z : complex variable
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A&S Ref:
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4.5.63
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Permalink:
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http://dlmf.nist.gov/4.33.E2
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Encodings:
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Annotations for §4.33 and Ch.4
4.33.3 tanh â¡ z = z â z3 3 + 2 15 ⢠z5 â 17 315 ⢠z7 + ⯠+ 2 2 ⢠n ⢠( 2 2 ⢠n â 1 ) ⢠B 2 ⢠n ( 2 ⢠n ) ! ⢠z 2 ⢠n â 1 + ⯠, | z | < 1 2 â¢ Ï . â
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Symbols:
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B n : Bernoulli numbers, Ï : the ratio of the circumference of a circle to its diameter, ! : factorial (as in n ! ), tanh â¡ z : hyperbolic tangent function, n : integer and z : complex variable
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A&S Ref:
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4.5.64
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Permalink:
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http://dlmf.nist.gov/4.33.E3
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Encodings:
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TeX, pMML, png
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See also:
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Annotations for §4.33 and Ch.4
For B 2 ⢠n see §24.2(i). For expansions that correspond to (4.19.4)â(4.19.9), change z to i ⢠z and use (4.28.8)â(4.28.13).
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