versity ni o U f
DEO PAT- Hyperbolic Functions ET RIE
S 2 2 a n The trigonometric functions cos α and cos α are defined using the unit circle x +y = 1 sk a a ew by measuring the distance α in the counter-clockwise direction along the circumference tch α of the circle. The area of the sector so determined is , so we can equivalently say that 2 cos α and cos α are derived from the unit circle x2 + y2 = 1 by measuring off a sector α (shaded red )of area . The other four trigonometric functions can then be defined in 2 terms of cos and sin. Similarly, we may define hyperbolic functions cosh α and sinh α from the “unit hy- perbola” α x2 −y2 = 1 by measuring off a sector (shaded red )of area to obtain a point P whose 2 x- and y- coordinates are defined to be cosh α and sinh α.
y y
(cosh α, sinh α)
x x
Since at this point we do not yet know how to compute the areas of most curved regions, we must take it on faith that the six hyperbolic functions may be expressed simply in terms of the exponential function: eα − e−α eα + e−α sinh α = cosh α = 2 2
sinh α eα − e−α cosh α eα + e−α tanh α = = cotanh α = = cosh α eα + e−α sinh α eα − e−α
1 2 1 2 sech α = = cosech α = = cosh α eα + e−α sinh α eα − e−α
Note that the domains of sinh, cosh, tanh, and sech are (−∞, ∞) and the domains of cotanh and cosech are (−∞, 0) ∪ (0, ∞).
1 eα + e−α eα − e−α We can check that the point , lies on the unit hyperbola: 2 2 eα + e−α 2 eα − e−α 2 e2α + 2 + e−2α e2α − 2 + e−2α 4 − = − = = 1 2 2 4 4 4
“Pythagorean” Identities
This gives us the first important hyperbolic function identity:
cosh2 α − sinh2 α ≡ 1
This may be used to derive two other identities relating the two other pairs of hyperbolic functions: 1 − tanh2 α = sech 2α and cotanh 2α − 1 = cosech 2α
Odd and Even Identities It is clear that sinh, tanh, cotanh xand cosech are odd functions, while cosh, cotanh , and sech are even, so we have the corresponding identities: sinh(−x) =−sinh x, tanh(−x) =−tanh x, cotanh (−x) =−cotanh x, cosech (−x) =−cosech x cosh(−x) = cosh x, sech (−x) = sech x.
Sum and Difference Identities We can use the above formulas for the hyperbolic functions in terms of ex to derive analogs of the identities for the trigonometric functions:
eα − e−α eβ + e−β (eα − e−α)(eβ + e−β) sinh α cosh β = = = 2 2 4 eα+β + eα−β − e−α+β − e−α−β 4
eβ − e−β eα + e−α (eβ − e−β)(eα + e−α) sinh β cosh α = = = 2 2 4
2 eβ+α + eβ−α − e−β+α − e−β−α 4
Adding these two products gives: sinh α cosh β + sinh β cosh α = eα+β + eα−β − e−α+β − e−α−β eβ+α + eβ−α + e−β+α − e−β−α + = 4 4
2eα+β − 2e−α−β eα+β − e−α−β e(α+β) − e−(α+β) = = = sinh(α + β) 4 2 2
and subtracting these two products gives: sinh α cosh β − sinh β cosh α = eα+β + eα−β − e−α+β − e−α−β eβ+α + eβ−α + e−β+α − e−β−α − = 4 4
2eα−β − 2e−(α−β) eα−β − e−(α−β) = = sinh(α − β) 4 2
Similarly, eα + e−α eβ + e−β (eα + e−α)(eβ + e−β) cosh α cosh β = = = 2 2 4 eα+β + eα−β + eβ−α + e−α−β 4
eα − e−α eβ − e−β (eα − e−α)(eβ − e−β) sinh α sinh β = = = 2 2 4 eα+β − eα−β − eβ−α + e−α−β 4
Adding these two products gives cosh α cosh β + sinh α sinh β = eα+β + eα−β + eβ−α + e−α−β eα+β − eα−β − eβ−α + e−α−β + = 4 4
3 2eα+β + 2e−α−β eα+β + e−(α+β) = = cosh(α + β) 4 2
and subtracting them gives: cosh α cosh β − sinh α sinh β = eα+β + eα−β + eβ−α + e−α−β eα+β − eα−β − eβ−α + e−α−β − = 4 4
2eα−β + 2e−α+β eα−β + e−(α−β) = = cosh(α − β) 4 2
Summarizing, we have four identities: sinh(α + β) ≡ sinh α cosh β + sinh β cosh α sinh(α − β) ≡ sinh α cosh β − sinh β cosh α cosh(α + β) ≡ cosh α cosh β + sinh α sinh β cosh(α − β) ≡ cosh α cosh β − sinh α sinh β which are almost exactly parallel to those for the trigonometric functions and may be used to derive sum and difference formulas for the other four hyperbolic functions.
Double and Half-“Angle” Identities
Letting β = α, we get: sinh 2α ≡ 2 sinh α cosh α, cosh 2α ≡ cosh2 α + sinh2 α ≡ 1 + 2 sinh2 α ≡ 2 cosh2 α − 1, so cosh 2α + 1 cosh 2α − 1 cosh2 α = and sinh2 α = , and thus: 2 2 cosh 2α + 1 cosh 2α − 1 cosh α = and sinh α = 2 2
4 α cosh α + 1 α cosh α − 1 cosh = and sinh = 2 2 2 2
Derivatives d d ex − e−x ex − (−e−x) ex + e−x (sinh x) = = = = cosh x dx dx 2 2 2 d d ex + e−x ex + (−e−x) ex − e−x (cosh x) = = = = sinh x dx dx 2 2 2