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Gudermannian function

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1: 4.46 Tables
§4.46 Tables
2: 19.10 Relations to Other Functions
In each case when y = 1 , the quantity multiplying R C supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. For relations to the Gudermannian function gd ( x ) and its inverse gd - 1 ( x ) 4.23(viii)), see (19.6.8) and
3: 19.6 Special Cases
4: 4.23 Inverse Trigonometric Functions
§4.23(viii) Gudermannian Function
4.23.39 gd ( x ) = 0 x sech t d t , - < x < .
The inverse Gudermannian function is given by
4.23.41 gd - 1 ( x ) = 0 x sec t d t , - 1 2 π < x < 1 2 π .
5: 22.16 Related Functions
22.16.4 am ( x , 0 ) = x ,
22.16.5 am ( x , 1 ) = gd ( x ) .
For the Gudermannian function gd ( x ) see §4.23(viii). …
6: 4.40 Integrals
4.40.5 sech x d x = gd ( x ) .
7: 4.26 Integrals
4.26.5 sec x d x = gd - 1 ( x ) , - 1 2 π < x < 1 2 π .
8: 19.9 Inequalities
9: 22.1 Special Notation
The notation sn ( z , k ) , cn ( z , k ) , dn ( z , k ) is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). …