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Showing content from https://dlmf.nist.gov/4.22 below:

§4.22 Infinite Products and Partial Fractions ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.22 Infinite Products and Partial Fractions ⓘ
Keywords:
infinite products, partial fractions, trigonometric functions
Notes:
See Hobson (1928, Chapter 17), Levinson and Redheffer (1970, pp. 387–389).
Referenced by:
§20.5(iii), §4.36
Permalink:
http://dlmf.nist.gov/4.22
See also:
Annotations for Ch.4
4.22.1 sin ⁡ z = z ⁢ ∏ n = 1 ∞ ( 1 − z2 n2 ⁢ π2 ) , ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, sin ⁡ z : sine function, n : integer and z : complex variable
A&S Ref:
4.3.89
Permalink:
http://dlmf.nist.gov/4.22.E1
Encodings:
TeX, pMML, png
See also:
Annotations for §4.22 and Ch.4
4.22.2 cos ⁡ z = ∏ n = 1 ∞ ( 1 − 4 ⁢ z2 ( 2 ⁢ n − 1 ) 2 ⁢ π2 ) . ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, cos ⁡ z : cosine function, n : integer and z : complex variable
A&S Ref:
4.3.90
Permalink:
http://dlmf.nist.gov/4.22.E2
Encodings:
TeX, pMML, png
See also:
Annotations for §4.22 and Ch.4

When z ≠ n ⁢ π , n ∈ ℤ ,

4.22.3 cot ⁡ z = 1 z + 2 ⁢ z ⁢ ∑ n = 1 ∞ 1 z2 − n2 ⁢ π2 , ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, cot ⁡ z : cotangent function, n : integer and z : complex variable
A&S Ref:
4.3.91
Referenced by:
§22.12
Permalink:
http://dlmf.nist.gov/4.22.E3
Encodings:
TeX, pMML, png
See also:
Annotations for §4.22 and Ch.4
4.22.4 csc2 ⁡ z = ∑ n = − ∞ ∞ 1 ( z − n ⁢ π ) 2 , ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, csc ⁡ z : cosecant function, n : integer and z : complex variable
A&S Ref:
4.3.92
Permalink:
http://dlmf.nist.gov/4.22.E4
Encodings:
TeX, pMML, png
See also:
Annotations for §4.22 and Ch.4
4.22.5 csc ⁡ z = 1 z + 2 ⁢ z ⁢ ∑ n = 1 ∞ ( − 1 ) n z2 − n2 ⁢ π2 . ⓘ
Symbols:
π : the ratio of the circumference of a circle to its diameter, csc ⁡ z : cosecant function, n : integer and z : complex variable
A&S Ref:
4.3.93
Referenced by:
§22.12
Permalink:
http://dlmf.nist.gov/4.22.E5
Encodings:
TeX, pMML, png
See also:
Annotations for §4.22 and Ch.4

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