HYPERBOLIC FUNCTIONS
12. HYPERBOLIC FUNCTIONS
Let us look at sin z and cos z for pure imaginary z, that is, z = iy:
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
−e −z ,
sinh z =
e z +e −z
cosh z =
The other hyperbolic functions are named and defined in a similar way to parallel the trigonometric functions:
sinh z
tanh z =
coth z =
cosh z
tanh z
sech z =
csch z =
cosh z
sinh z
(See Problem 38 for the reason behind the term “hyperbolic” functions.) We can write (12.1) as
sin iy = i sinh y,
cos iy = cosh y.
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 z. Because of this relation between hyperbolic and trigonometric func- 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).
cosh 2 2 z=1 (compare sin 2 z + cos z − sinh 2 z = 1),
cos z = − sin z). dz
cosh z = sinh z
(compare
dz
PROBLEMS, SECTION 12
Verify each of the following by using equations (11.4), (12.2), and (12.3). 1. sin z = sin(x + iy) = sin x cosh y + i cos x sinh y
Section 12 Hyperbolic Functions
2. cos z = cos x cosh y − i sin x sinh y 3. sinh z = sinh x cos y + i cosh x sin y 4. cosh z = cosh x cos y + i sinh x sin y
5. sin 2z = 2 sin z cos z 6. cos 2z = cos 2 z − sin 2 z
7. sinh 2z = 2 sinh z cosh z 2 2 8. d cosh 2z = cosh z + sinh z 9.
dz cos z = − sin z 10. d cosh z = sinh z
11. cosh 2 dz 2 z − sinh z=1 4 4 12. 1 cos z + sin
z=1− 2 sin 2z 2 13. cos 3z = 4 cos 3 z − 3 cos z 14. sin iz = i sinh z
15. sinh iz = i sin z 16. tan iz = i tanh z
17. tanh iz = i tan z
tan x + i tanh y
18. tan z = tan(x + iy) =
1 − i tan x tanh y
tanh x + i tan y 19. tanh z = 1 + i tanh x tan y
20. Show that e nz = (cosh z + sinh z) n = cosh nz + sinh nz. Use this and a similar equation for e −nz to find formulas for cosh 3z and sinh 3z in terms of sinh z and cosh z.
21. Use a computer to plot graphs of sinh x, cosh x, and tanh x. 22. Using (12.2) and (8.1), find, in summation form, the power series for sinh x and
cosh x. Check the first few terms of your series by computer. Find the real part, the imaginary part, and the absolute value of
25. sin(x − iy) 26. cosh(2 − 3i)
23. cosh(ix)
24. cos(ix)
27. sin(4 + 3i)
28. tanh(1 − iπ)
Find each of the following in the x + iy form and check your answers by computer. „
3πi « iπ 29. cosh 2π i
30. tanh
31. sinh ln 2 + 4 3
„ iπ « 32. cosh
35. cosh(iπ + 2)
36. sinh 1+ iπ
37. cos(iπ)
38. 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
x = cos t, y = sin t, satisfy the equation of a circle x 2 +y 2 = 1, while x = cosh t, y = sinh t, satisfy the equation of a hyperbola x 2 −y 2 = 1.
72 Complex Numbers Chapter 2
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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