Trigonometric functions, relations

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

The trigonometric functions developed in the previous seefidtiare lef pd to as circidar functions, as they are related to the circle shown in Fig, 11, somewhat less familiar family of functions, the hyperbolic funefens, c also be derived from the exponential. They are analogous to the circular iom considered above and can be defined bv the relations... 

Even in the case where k =, , , -- a general solution of the type given in Eq. (146) can be found. In fact, this case is of particular importance in many physical problems, as these Bessel functions are closely related to the ordinary trigonometric functions. 

Using these formulae it can be shown that when n is half an odd integer, e.g. I + then Jn(x) takes a particularly simple form and is related to trigonometric functions. By definition, for instance... 

Since negative angles arise when using trigonometric functions, it is important to establish how, for example, sin(-0) is related to sin0. The periodicity of the sine function yields the equality ... 

In the box, two additional methods to obtain Bessel functions are summarized. The generating function relates Bessel functions to the exponential, Spiegel, 1971[3]. This relation is useful for obtaining properties of the Bessel function for integral n. Recursions of the Bessel functions are generally derived this way. Bessel s integral relates Bessel and trigonometric function. 

On the other hand, the comments made for the case of nonutilization of an auxiliary elastic element in relation to the different types of response are also pertinent in the present context. In fact, if the trigonometric functions appearing in Eq. (17.293) are approximated by... 

Related Functions SIN, other trigonometric functions ATAN... 

Related Functions ATAN2, TAN, other trigonometric functions... 

Related Functions ATAN, ATAN2, other trigonometric functions TEXT... 

The remaining four hyperbolic functions, analogous to the remaining four trigonometrical functions, are tanh u, cosech u, sech u and coth u. Values for each of these functions may be deduced from their relations with sinh u and cosh u. Thus,... 

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