Hyperbolic sine and hyperbolic cosine are mathematical hyperbolic functions , also called hyperbolic sine and hyperbolic cosine ; they carry the symbols or , in older sources also and . The hyperbolic cosine describes, among other things, the course of a rope suspended at two points. Its graph is therefore also called a catenoid (chain line).
![\ sinh](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6649e190c30e91b280cc02b27acdfe00055e58)
![\ cosh](https://wikimedia.org/api/rest_v1/media/math/render/svg/011ccafd311d38f9bbc0fcf3f7b13f7d81469d84)
![{\ mathfrak {Sin}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fd20b274f4963f452baff4314cb7f32066fc26f)
![{\ mathfrak {Cos}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c113aba171e9a3171684438d173fef07b6033d06)
Definitions
![\ sinh x = {\ frac {1} {2}} \ left (e ^ {x} -e ^ {- x} \ right) = - i \, \ sin (i \, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ab79a127a462164ee61aa5c41127d0e11c44c44)
![\ cosh x = {\ frac {1} {2}} \ left (e ^ {x} + e ^ {- x} \ right) = \ cos (i \, x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e46c1ec79b353fe76fc6914037adb05b2f8b15f5)
The functions sinh and cosh are the odd or even part of the exponential function ( ).
![{\ displaystyle \ exp x = \ cosh x + \ sinh x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31f8912e23b183048ca1e4f75a8bdd12237434e6)
properties
Hyperbolic sine (red) and hyperbolic cosine (blue) for real x.
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Hyperbolic sine
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Hyperbolic cosine
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Domain of definition
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Range of values
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periodicity
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no
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no
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monotony
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strictly monotonously increasing
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strictly monotonically decreasing strictly monotonically increasing
![0 \ leq x <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/57153c7e9e79417412ae177b741678396de46b5f) |
Symmetries
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Point symmetry to the origin
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Axial symmetry to the ordinate
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Asymptotic functions
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zeropoint
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no
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Jump points
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no
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no
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Poles
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no
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no
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Extremes
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no
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Minimum at
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Turning points
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no
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Special values
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with the golden ratio
For the hyperbolic cosine the following applies in particular:
![\ int \ limits _ {{- \ infty}} ^ {\ infty} {\ frac {{\ mathrm d} x} {\ cosh x}} = \ pi.](https://wikimedia.org/api/rest_v1/media/math/render/svg/de71849ef93b0546eea405e00ee96d2e8bdb7a4b)
Inverse functions
The hyperbolic sine is bijective on , and therefore has an inverse function , which one area hyperbolic sine calls.
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
The hyperbolic cosine forms the interval bijectively on the interval and can therefore be inverted to a limited extent. The inverse function of this is called the hyperbolic areakosine![[0, + \ infty [](https://wikimedia.org/api/rest_v1/media/math/render/svg/207d226525287a9b2ebb3ba52c61454a0df207b2)
![[1, + \ infty [](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcfce3eaa1ee5c0bc94b8128b4db0b7ef47569b7)
Both inverse functions, hyperbolic area and hyperbolic area, can be calculated with the help of more elementary functions as follows:
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Derivatives
The derivative of the hyperbolic sine is the hyperbolic cosine and the derivative of the hyperbolic cosine is the hyperbolic sine:
![{\ begin {aligned} {\ frac {{\ mathrm {d}}} {{\ mathrm {d}} x}} \ sinh x & = \ cosh x \\ {\ frac {{\ mathrm {d}}} {{\ mathrm {d}} x}} \ cosh x & = \ sinh x \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4746ad2662b08b6a75fcc7bde97915af542bd201)
Antiderivatives
![{\ displaystyle {\ begin {aligned} \ int \ sinh x \, \ mathrm {d} x & = \ cosh x + C \\\ int \ cosh x \, \ mathrm {d} x & = \ sinh x + C \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e800b16700ef05475644e10af6b1eb84cb5b4d3)
Relationships (between the two functions and others)
![\ cosh ^ {2} x \! \; - \ sinh ^ {2} x = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/c282ef1ab4ba82ace18a1dc64271b53ad6488916)
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( Euler's identity )
![\ cosh ({{\ rm {arsinh}}} (x)) = {\ sqrt {x ^ {2} +1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cec55a08745f2399d38e13209415f5f7f8608c)
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( Hyperbolic equation )
Addition theorems
![{\ begin {aligned} \ sinh (x \ pm y) & = \ sinh x \ cosh y \ pm \ cosh x \ sinh y \\\ cosh (x \ pm y) & = \ cosh x \ cosh y \ pm \ sinh x \ sinh y \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af9b27d216f7205f270bafc0e5fa0ec7adc810cb)
in particular applies to :
![y: = x](https://wikimedia.org/api/rest_v1/media/math/render/svg/8eabd7a86b1628cca397c766b77d31278ecb8d52)
![{\ begin {aligned} \ sinh 2x & = 2 \ cdot \ sinh x \ cosh x \ \\\ cosh 2x & = \ cosh ^ {2} x + \ sinh ^ {2} x = 2 \ cdot \ cosh ^ {2} x-1 = 2 \ cdot \ sinh ^ {2} x + 1 \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f1b8f4e4680c6b0f21397b62123af2759ea940)
and for :
![y: = 2x](https://wikimedia.org/api/rest_v1/media/math/render/svg/29aee52bd727fff75cafeb65a9f741aaf37e3fee)
![{\ begin {aligned} \ sinh 3x & = 4 \ cdot \ sinh ^ {3} x + 3 \ sinh x \ \\\ cosh 3x & = 4 \ cdot \ cosh ^ {3} x-3 \ cosh x \ end { aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09e214b31ef27aaa37440ce93a20af2f75a5afe)
Molecular formulas
![{\ begin {aligned} \ sinh x \ pm \ sinh y & = 2 \ sinh {\ frac {x \ pm y} 2} \ cosh {\ frac {x \ mp y} 2} \\\ cosh x + \ cosh y & = 2 \ cosh {\ frac {x + y} 2} \ cosh {\ frac {xy} 2} \\\ cosh x- \ cosh y & = 2 \ sinh {\ frac {x + y} 2} \ sinh { \ frac {xy} 2} \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc12b916c1d1efed84174cc45204e85964677aa9)
Series developments
The Taylor series of the hyperbolic sine or hyperbolic cosine with the development point is:
![x = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc)
![{\ begin {aligned} \ sinh x & = \ sum _ {{n = 0}} ^ {\ infty} {\ frac {x ^ {{{2n + 1}}} {(2n + 1)!}} = x + {\ frac {x ^ {3}} {3!}} + {\ frac {x ^ {5}} {5!}} + \ dotsb \\\ cosh x & = \ sum _ {{n = 0}} ^ {\ infty} {\ frac {x ^ {{2n}}} {(2n)!}} = 1 + {\ frac {x ^ {2}} {2!}} + {\ frac {x ^ { 4}} {4!}} + \ Dotsb \ end {aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/969043680f06aa6c73e671e2832592ab6145822c)
Product developments
![{\ displaystyle {\ begin {aligned} & \ sinh x = x \ cdot \ prod _ {k = 1} ^ {\ infty} \ left (1 + {\ frac {x ^ {2}} {(k \ pi ) ^ {2}}} \ right) \ qquad \ qquad \ quad \\ & \ sinh {(\ pi x)} = \ pi x \ cdot \ prod _ {k = 1} ^ {\ infty} \ left ( 1 + {\ frac {x ^ {2}} {k ^ {2}}} \ right) \\ & \ cosh x = \ prod _ {k = 1} ^ {\ infty} \ left (1 + {\ frac {4x ^ {2}} {(2k-1) ^ {2} \ pi ^ {2}}} \ right) \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/107f3a008ac6095a97540ce00ea1058334f6076d)
Multiplication formulas
Be . Then for all complex :
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle {\ begin {aligned} & \ sinh z = {\ left ({\ frac {2} {i}} \ right)} ^ {\! \! n-1} \, \ prod \ limits _ { k = 0} ^ {n-1} \ sinh {\ frac {z + k \, \ pi \, i} {n}} \\ & \ cosh z = 2 ^ {n-1} \ prod \ limits _ {k = 0} ^ {n-1} \ cosh {\ frac {z + \ left (k - {\ frac {n-1} {2}} \ right) \, \ pi \, i} {n}} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f5b4487db445e6a6982379afef1f7a1bd9d9a2)
Complex arguments
With applies:
![x, y \ in \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc)
![{\ displaystyle {\ begin {aligned} \ sinh (x + i \, y) & = \ cos y \, \ sinh x + i \ sin y \, \ cosh x \\\ cosh (x + i \, y ) & = \ cos y \, \ cosh x + i \ sin y \, \ sinh x \\\ sin (x + i \, y) & = \ sin x \, \ cosh y + i \ cos x \, \ sinh y \\\ cos (x + i \, y) & = \ cos x \, \ cosh yi \ sin x \, \ sinh y \\\ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c06f5eb970d99d7106170087e92ba6110e822527)
For example, the third and fourth equations follow in the following way:
With applies
![z = x + i \, y](https://wikimedia.org/api/rest_v1/media/math/render/svg/db45cab0be65d6a2f49fc11df10504bf3d5cfba8)
By comparison of coefficients it follows:
Applications
Solution of a differential equation
The function
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With
solves the differential equation
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Chain line
A homogeneous rope that only sags due to its own weight can be described by a hyperbolic cosine function. Such a curve is also called a chain line, chain curve or catenoid .
Lorentz transformation
With the help of the rapidity , the transformation matrix for a special Lorentz transformation (also Lorentz boost ) in the x direction can be represented as follows (for transformations in other directions there are similar matrices):
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![L = {\ begin {pmatrix} \ cosh \ lambda & - \ sinh \ lambda & 0 & 0 \\ - \ sinh \ lambda & \ cosh \ lambda & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \ end {pmatrix}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a962240cb30f30965d98c6523ad9fbf0dd14c9a8)
You can see a great similarity to rotary dies ; So one can easily see the analogy between special Lorentz transformations in four-dimensional space - time and rotations in three-dimensional space.
cosmology
The hyperbolic sine also occurs in cosmology . The evolution of the scale factor over time in a flat universe containing essentially only matter and dark energy (which is a good model for our actual universe) is described by
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in which
![{\ displaystyle t _ {\ mathrm {ch}} = {\ frac {2} {3 {\ sqrt {\ Omega _ {\ Lambda, 0}}} H_ {0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/838197ac417a79b94151c83213c6df55557eb35d)
is a characteristic timescale. is the current value of the Hubble parameter, the density parameter for dark energy. The derivation of this result can be found in the Friedmann equations . In the case of the time dependence of the density parameter of matter, on the other hand, the hyperbolic cosine occurs:
![H_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43910602a221b7a4c373791f94793e3008622070)
![\ Omega _ {{\ Lambda, 0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14cab36840b3568cb20ff85ea6dc0de941232f16)
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See also
Web links
Individual evidence
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↑ Dr. Franz Brzoska, Walter Bartsch: Mathematical formula collection . 2nd improved edition. Fachbuchverlag Leipzig, 1956.