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