Chapter 2 70 Complex Numbers 12. HYPERBOLIC FUNCTIONS Let us look at sinz and cos z for pure imaginary , that is, iy: Z. e"V -e sin iy = i 2i 2 (12.1) e e e+e cos iy= - 2 The real functions on the right have special names because these particular combi- nations of exponentials arise frequently in problems. They are called the hyperbolic sine (abbreviated sinh) and the hyperbolic cosine (abbreviated cosh). Their defini- tions for all z are sinh z (12.2) e e cosh 2 The other hyperbolic functions are named and defined in a similar way to parallel the trigonometric functions: sinh tanh z= cosh cothz tanh (12.3) sech z cschz cosh sinh (See Problem 38 for the reason behind the term "hyperbolic" functions.) We can write (12.1) as sin iy=i sinh y (12.4) cos iy coshy. Then we see that the hyperbolic functions of y are (except for one i factor) the trigonometric functions of iy. From (12.2) we can show that (12.4) holds with y replaced by . Because of this relation between hyperbolic and trigonometric fune tions, the formulas for hyperbolic functions look very much like the corresponding trigonometric identities and calculus formulas. They are not identical, however. Example. You can prove the following formulas (see Problems 9, 10, 11 and 38) cosh2-sinh 1 (compare sinz + cos z = 1 ), d cosh zsinh z dz d sin ). (comparecos z dz PROBLEMS, SECTION 12 Verify each of the following by using equations (11.4), (12.2), and (12.3). sinzsin(a+iy) = sin cosh y + i cosz sinhy 1. Hyperbolic Functions 71 Section 12 COs zcos cosh y-i sin z sinh w sinh z=sinhz cos y + i cosh z sin y 2. 3. 4. coshz= cosh cos y +i sinh r sin y sin 2z2 sin z cos z 5. cOs 22cOs: - sin 7. sinh 2z 2 sinh z cosh 6. d cosh 2z cosh2 zsinh2 COs z dz 9. sin 2 8. = cosh2z-sinh2z = 1 (compare sin z + cos? z = 1), d Cosh z= sinh d (compare cos z-sinz). sin 2) dz PROBLEMS, SECTION 12 Verify each of the following by using equations (11.4), (12.2), and (12.3). 1. sin zsin(riy)sin r cosh y+i cos r sinh y Hyperbolic Functions 71 Section 12 sinh zsinhr cos y +i cosh z sin y Cos zcos r cosh y-i sin z sinh y 2. 3. Cosh z= cosh z cos y +i sinh r sin y 4. sin 2z2 sin z cos z 5. Cos 2z cos2- sin sinh 2z 2 sinh z cosh z 6. d COs 2 sin dz cosh 2:= COsh+ sinh2 2+ sinh2 9. 8. d Cosh z= sinhz P 11. cosh-sinh2 z = 1 10. Cos sin : = 1 --sin 22 cos 3z=4 cos3 z-3 cos z 12. 13. 15. sinh izi sin z sin iz=i sinh 14. tanhizi tan z tan iz=itanh 17. 16. tan zi tanh y tanz= tan( +y)= itanrtanhy 18. tanhri tan y 1+i tanhr tany tanh z 19. Show that ez = (cosh zsinh z) = coshnz sinh nz. Use this and a similar equation for e- to find formulas for cosh 3z and sinh 3z in terms of sinh z and cosh 20. Use a computer to plot graphs of sinh r, cosh z, and tanh z. 21 22. Using (12.2) and (8.1), find, in summation form, the power series for sinh z and Cosh . Check the first few terms of your series by computer Find the real part, the imaginary part, and the absolute value of 23. соsh(ix) 24. сок(ir) 25. sin(-iy) tanh(1 i) 26. соsh(2 -3) 27. sin(4+3i) 28. Find each of the following in the r+iy form and check your answers by computer. Злі tanh sinh In 2+ cosh 2i 31. 29. 30. ( (-) in cosh sin 32 In 3 33 tan i 34 37. сов (iя) 35. cosh(i+2) sinh 1 36. The functions sin t, cos t,are called "circular functions" and the functions sinh t, cosh t, are called "hyperbolic functions". To see a reason for this, show that zcost, y sin t, satisfy the equation of a circle 2 y= 1, while z = cosh t, y=sinh t, satisfy the equation of a hyperbola z2-y= 1