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General Identities

General Mathematical Identities for Analytic Functions

General Identities

Relations with other functions

With inverse function

This property is the definition of the inverse function and can hold without additional restrictions on (like , where is not ) for many named functions. In these situations, is in most cases free of branch cuts. For example, ; here means with , that is, the inverse sine function (do not confuse this with the reciprocal function ).

Some of the functions are invertible: their inversions can coincide with the original , but for other values of the parameters. For example, the inverse function for the power function is also the power function , and the relation takes place only under the restriction . In general cases the following relation takes place: .

The last property for the inverse function of the direct function can be valid under special restrictions for (where typically is not ). For example,

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