Functions circular

Because of this duality, every relation involving circular functions btf formal counterpart in foe corresponding hyperbolic functions, and vice ... 

Thus, the various relations between the hyperbolic functions can be derived as carried out above for the circular functions. For example,... 

On substituting Eq. (5.43) into Eq. (5.41) we have, using the orthogonality properties of the circular functions,... 

The value of a is always negative, so that the solutions will always be oscillatory and expressible in terms of the normal circular functions, with a frequency equal to the square root of — a, namely... 

This is called the circular measure of an angle and, for this reason, trigonometrical functions are sometimes called circular functions. This property is possessed by no plane curve other than the circle. For instance, the hyperbola, though symmetrically placed with respect to its centre, is not at all points equidistant from it. The same thing is true of the ellipse. The parabola has no centre. 

Unlike the circular functions, the ratios x/a, yja, when referred to the hyperbola, do not represent angles. An hyperbolic function expresses a certain relation between the coordinates of a given portion on the arc of a rectangular hyperbola. 

The Shift and Hyp button make the inverse circular functions (Fig. 10.17) and the hyperbolic functions (Fig. 10.18), respectively, available. 

To avoid obscure steps in this introduction, even the imaginary number is brought logically (i.e., demonstrated) as a useful factor for combining variables. For the same reason, the exponential function is entirely defined from its basic properties and harmonic (circular) functions are deduced. No prerequisite about these elementary mathematical notions is therefore required for understanding what a wave function is really. 

Although accurate values of the optical properties of magnetic lenses can be obtained only by numerical methods, in which the field distribution is first calculated by one of the various techniques available—finite differences, finite elements, and boundary elements in particular—their variation can be studied with the aid of field models. The most useful (though not the most accurate) of these is Glaser s bell-shaped model, which has the merits of simplicity, reasonable accuracy, and, above all, the possibility of expressing all the optical quantities such as focal length, focal distance, the spherical and chromatic aberration coefficients Cg and Q, and indeed all the third-order aberration coefficients, in closed form, in terms of circular functions. In this model, 5(z) is represented by... 

The dissimilarity D(PQ) between fragments P and Q is assessed using the Minkowski metric (equation 2 above), where V(P ) and V(Q ) are the Jth torsion angles for fragments P and Q, and summation is from J = 1 — Nt. Because torsion angles are circular functions, there is a phase restriction of -180< y<180, and the torsional dissimilarity measure must be expressed as ... 

This provided us with an extremely fast log function But when we needed more accurate results, we used other modification routines (though we compromized on execution time). Similar techniques have been employed, (especially) for circular functions. 

Circular Functions. A circle is defined as die locus of all points in a plane that are at a constant distance from a fixed point. Circles are described by the equation... 

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