Handbook of mathematics for engineers and scienteists part 11

The line connecting the midpoints of two sides of a triangle is called a midline of the triangle. The midline is parallel to and half as long as the third side (Fig. ). Let rj, b, and c be the lengths of the sides of a triangle; let Q, /?, and 7 be the respective opposite angles (Fig. 3Aa): let B and r be the circumradius and the inradius, respectively; and let p = -j(a +h + c) be the semiperimeter. | 38 Elementary Functions . Hyperbolic functions of multiple argument. cosh 2x 2 cosh2 x -1 sinh 2x 2 sinh x cosh x cosh 3x -3 cosh x 4 cosh3 x sinh 3x 3 sinh x 4 sinh3 x cosh 4x 1 - 8 cosh2 x 8 cosh4 x sinh4x 4 cosh x sinh x 2 sinh3 x cosh 5x 5 cosh x -20 cosh3 x 16 cosh5 x sinh 5x 5 sinh x 20 sinh3 x 16 sinh5 x. 2 z fe 1 cosh nx 2 -1 cosh x --------C - 22 -2k-2 cosh x -2k-2 2 k 1 -k-2 k 0 -1 2 sinh nx sinh x 2 -k-1 C -k-1 cosh x -2k-1. k 0 Here C are binomial coefficients and A stands for the integer part of the number A. . Hyperbolic functions of half argument. sinh x sign J . 1 x I cosh x 1 cosh A -------- 2 V 2 x sinh x cosh x - 1 tanh r -T--- 2 cosh x 1 sinh x x sinh x C H1 - z 1 2 cosh x - 1 cosh x 1 sinh x . Differentiation formulas. d sinh x d cosh x sinh x dx dx d tanh x 1 d coth x 1 dx cosh2 x dx sinh2 x . Integration formulas. sinh x dx cosh x C cosh x dx sinh x C tanh x dx ln cosh x C coth x dx ln sinh x C where C is an arbitrary constant. . Inverse Hyperbolic Functions 39 . Expansion in power series. 2 -4 -6 -2n cp rp - ÍỴ n 1 -1 . x . x . x . x cosh x I x2n 1 ------ 2n 1 z 2 22n - B x 1 ------ 7---------- 2n 22n B2n x2n 1 -------------- 3 5 7 zy -1 zy zy XXX sinh x x x3 2x5 17x7 anh x x - T 55 - 515 35 coth x 757T---- 1 n-1 x 3 45 945 2n where Bn are Bernoulli numbers see Subsection . x to x to x n 2 x n . Relationship with trigonometric functions. sinh ix i sin x cosh ix cos x tanh ix i tan x coth ix i cot x i2 1. . Inverse Hyperbolic Functions . Definitions. Graphs of Inverse Hyperbolic Functions . Definitions of inverse hyperbolic functions. Inverse hyperbolic junctions are the functions that are inverse to hyperbolic functions. The following notation is used for inverse hyperbolic functions arcsinh x sinh-1 x arccosh x cosh-1 x arctanh x tanh-1 x arccoth x coth-1 x inverse of hyperbolic sine inverse of hyperbolic cosine inverse of hyperbolic tangent inverse of hyperbolic .