Hyperbolic functions

Error creating thumbnail: About to transcode 1 SVG file(s) Converting Sinh_cosh_tanh.svg to /var/www/html/w/images/temp/transform_00ff71ed10cc.png ... org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_e9407a1db5dc3298a814cf7e/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_e9407a1db5dc3298a814cf7e/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_e9407a1db5dc3298a814cf7e/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_e9407a1db5dc3298a814cf7e/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.apache.batik.transcoder.TranscoderException: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_e9407a1db5dc3298a814cf7e/Sinh_cosh_tanh.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:228) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) Caused by: org.apache.batik.bridge.BridgeException: file:/var/www/html/w/images/temp/svg_e9407a1db5dc3298a814cf7e/Sinh_cosh_tanh.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.bridge.BridgeContext.getReferencedNode(BridgeContext.java:762) at org.apache.batik.bridge.BridgeContext.getReferencedElement(BridgeContext.java:804) at org.apache.batik.bridge.SVGUseElementBridge.buildCompositeGraphicsNode(SVGUseElementBridge.java:124) at org.apache.batik.bridge.SVGUseElementBridge.createGraphicsNode(SVGUseElementBridge.java:98) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:213) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) ... 7 more ... error (SVGConverter.error.while.rasterizing.file)

Hyperbolic functions are different versions of trigonometric functions. They are defined using a hyperbola instead of a circle. Also, as the derivatives of $sin(t)$ and $cos(t)$ are $cos(t)$ and $-sin(t)$ , the derivatives of $sinh(t)$ and $cosh(t)$ are $cosh(t)$ and $sinh(t)$ .

The basic hyperbolic functions are:^[1]

hyperbolic sine " $sinh$ " (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),^[2]
hyperbolic cosine " $cosh$ " (/ˈkɒʃ, ˈkoʊʃ/),

Using these, you can get:^[3]

hyperbolic tangent " $tanh$ " (/ˈtæŋ, ˈtæntʃ, ˈθæn/),
hyperbolic cotangent " $coth$ " (/ˈkɒθ, ˈkoʊθ/),
hyperbolic secant " $sech$ " (/ˈsɛtʃ, ˈʃɛk/),
hyperbolic cosecant " $csch$ " or " $cosech$ " (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/)

and these have similar logic to the normal trigonometric functions.

The inverse hyperbolic functions are:

inverse hyperbolic sine " $arsinh$ " (also " $sinh -1$ ", " $asinh$ " or sometimes " $arcsinh$ ")^[4]^[5]
inverse hyperbolic cosine " $arcosh$ " (also " $cosh -1$ ", " $acosh$ " or sometimes " $arccosh$ ")
inverse hyperbolic tangent " $artanh$ " (also " $tanh -1$ ", " $atanh$ " or sometimes " $arctanh$ ")
inverse hyperbolic cotangent " $arcoth$ " (also " $coth -1$ ", " $acoth$ " or sometimes " $arccoth$ ")
inverse hyperbolic secant " $arsech$ " (also " $sech -1$ ", " $asech$ " or sometimes " $arcsech$ ")
inverse hyperbolic cosecant " $arcsch$ " (also " $arcosech$ ", " $csch -1$ ", " $cosech -1$ "," $acsch$ ", " $acosech$ ", or sometimes " $arccsch$ " or " $arccosech$ ")

Definitions

With hyperbolic angle u, the hyperbolic functions sinh and cosh can be defined using the exponential function e^u.^[1]^[3] In the figure [math]\displaystyle{ A =(e^{-u}, e^u), \ B=(e^u, \ e^{-u}), \ OA + OB = OC }[/math] .

Exponential definitions

sinh x

is half the difference of

e x

and

e - x

cosh x

is the average of

e x

and

e - x

Hyperbolic sine: the odd part of the exponential function, that is, [math]\displaystyle{ \sinh x = \frac {e^x - e^{-x}} {2} = \frac {e^{2x} - 1} {2e^x} = \frac {1 - e^{-2x}} {2e^{-x}}. }[/math]
Hyperbolic cosine: the even part of the exponential function, that is, [math]\displaystyle{ \cosh x = \frac {e^x + e^{-x}} {2} = \frac {e^{2x} + 1} {2e^x} = \frac {1 + e^{-2x}} {2e^{-x}}. }[/math]

sinh, cosh and tanh

csch, sech and coth

Hyperbolic tangent: [math]\displaystyle{ \tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1}. }[/math]
Hyperbolic cotangent: for $x \neq 0$ , [math]\displaystyle{ \coth x = \frac{\cosh x}{\sinh x} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1}. }[/math]
Hyperbolic secant: [math]\displaystyle{ \operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = \frac{2e^x} {e^{2x} + 1}. }[/math]
Hyperbolic cosecant: for $x \neq 0$ , [math]\displaystyle{ \operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = \frac{2e^x} {e^{2x} - 1}. }[/math]

Hyperbolic Functions Media

Error creating thumbnail: About to transcode 1 SVG file(s)
Converting Sinh_cosh_tanh.svg to /var/www/html/w/images/temp/transform_a8db7029780e.png ... org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_101925d2d46cb507217c66a1/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_101925d2d46cb507217c66a1/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_101925d2d46cb507217c66a1/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_101925d2d46cb507217c66a1/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.apache.batik.transcoder.TranscoderException: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_101925d2d46cb507217c66a1/Sinh_cosh_tanh.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:228) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) Caused by: org.apache.batik.bridge.BridgeException: file:/var/www/html/w/images/temp/svg_101925d2d46cb507217c66a1/Sinh_cosh_tanh.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.bridge.BridgeContext.getReferencedNode(BridgeContext.java:762) at org.apache.batik.bridge.BridgeContext.getReferencedElement(BridgeContext.java:804) at org.apache.batik.bridge.SVGUseElementBridge.buildCompositeGraphicsNode(SVGUseElementBridge.java:124) at org.apache.batik.bridge.SVGUseElementBridge.createGraphicsNode(SVGUseElementBridge.java:98) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:213) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) ... 7 more ... error (SVGConverter.error.while.rasterizing.file)

The hyperbolic sine (red), hyperbolic cosine (green) and hyperbolic tangent (blue) graphed on the same axes.*Replaces en:Image:sinh cosh tanh.png Instructions Generated with en:Matplotlib using the script below. Edited using en:Inkscape to fix clipping issues
$sinh x$ is half the difference of $ex$ and $e - x$
$cosh x$ is the average of $ex$ and $e - x$
Error creating thumbnail: About to transcode 1 SVG file(s)
Converting Sinh_cosh_tanh.svg to /var/www/html/w/images/temp/transform_04fa0fac29a5.png ... org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_a9743507d5ee5b395bec6257/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_a9743507d5ee5b395bec6257/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_a9743507d5ee5b395bec6257/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_a9743507d5ee5b395bec6257/Sinh_cosh_tanh.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.apache.batik.transcoder.TranscoderException: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_a9743507d5ee5b395bec6257/Sinh_cosh_tanh.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:228) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) Caused by: org.apache.batik.bridge.BridgeException: file:/var/www/html/w/images/temp/svg_a9743507d5ee5b395bec6257/Sinh_cosh_tanh.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.bridge.BridgeContext.getReferencedNode(BridgeContext.java:762) at org.apache.batik.bridge.BridgeContext.getReferencedElement(BridgeContext.java:804) at org.apache.batik.bridge.SVGUseElementBridge.buildCompositeGraphicsNode(SVGUseElementBridge.java:124) at org.apache.batik.bridge.SVGUseElementBridge.createGraphicsNode(SVGUseElementBridge.java:98) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:213) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) ... 7 more ... error (SVGConverter.error.while.rasterizing.file)

The hyperbolic sine (red), hyperbolic cosine (green) and hyperbolic tangent (blue) graphed on the same axes.*Replaces en:Image:sinh cosh tanh.png Instructions Generated with en:Matplotlib using the script below. Edited using en:Inkscape to fix clipping issues
Error creating thumbnail: About to transcode 1 SVG file(s)
Converting Csch_sech_coth.svg to /var/www/html/w/images/temp/transform_bc6f448d7724.png ... org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_c6771d5703466e08efc52b23/Csch_sech_coth.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_c6771d5703466e08efc52b23/Csch_sech_coth.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.w3c.dom.DOMException: file:/var/www/html/w/images/temp/svg_c6771d5703466e08efc52b23/Csch_sech_coth.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. at org.apache.batik.css.engine.CSSEngine.getCascadedStyleMap(CSSEngine.java:775) at org.apache.batik.css.engine.CSSEngine.getComputedStyle(CSSEngine.java:867) at org.apache.batik.bridge.CSSUtilities.getComputedStyle(CSSUtilities.java:81) at org.apache.batik.bridge.CSSUtilities.convertDisplay(CSSUtilities.java:563) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:206) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) ERROR: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_c6771d5703466e08efc52b23/Csch_sech_coth.svg: The attribute "stroke-miterlimit" represents an invalid CSS value ("1e5"). Original message: The "stroke-miterlimit" property does not support dimension values. org.apache.batik.transcoder.TranscoderException: null Enclosed Exception: file:/var/www/html/w/images/temp/svg_c6771d5703466e08efc52b23/Csch_sech_coth.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:228) at org.apache.batik.transcoder.image.ImageTranscoder.transcode(ImageTranscoder.java:92) at org.apache.batik.transcoder.XMLAbstractTranscoder.transcode(XMLAbstractTranscoder.java:142) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:158) at org.apache.batik.apps.rasterizer.SVGConverter.transcode(SVGConverter.java:1008) at org.apache.batik.apps.rasterizer.SVGConverter.execute(SVGConverter.java:719) at org.apache.batik.apps.rasterizer.Main.execute(Main.java:956) at org.apache.batik.apps.rasterizer.Main.main(Main.java:1009) Caused by: org.apache.batik.bridge.BridgeException: file:/var/www/html/w/images/temp/svg_c6771d5703466e08efc52b23/Csch_sech_coth.svg:-1 Cannot find the referenced element: "#BitstreamVeraSans-Roman-20" specified on the element <use> - may be a problem of 'id' at org.apache.batik.bridge.BridgeContext.getReferencedNode(BridgeContext.java:762) at org.apache.batik.bridge.BridgeContext.getReferencedElement(BridgeContext.java:804) at org.apache.batik.bridge.SVGUseElementBridge.buildCompositeGraphicsNode(SVGUseElementBridge.java:124) at org.apache.batik.bridge.SVGUseElementBridge.createGraphicsNode(SVGUseElementBridge.java:98) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:213) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.buildGraphicsNode(GVTBuilder.java:219) at org.apache.batik.bridge.GVTBuilder.buildComposite(GVTBuilder.java:171) at org.apache.batik.bridge.GVTBuilder.build(GVTBuilder.java:82) at org.apache.batik.transcoder.SVGAbstractTranscoder.transcode(SVGAbstractTranscoder.java:210) ... 7 more ... error (SVGConverter.error.while.rasterizing.file)

The hyperbolic cosecant (red), hyperbolic secant (green) and hyperbolic cotangent (blue) graphed on the same axes.*Replaces Csch sech coth.png. Instructions Generated with en:Matplotlib using the script below. Edited using en:Inkscape to fix clipping issues.
Circle and hyperbola tangent at $(1, 1)$ display geometry of circular functions in terms of circular sector area $u$ and hyperbolic functions depending on hyperbolic sector area $u$ .
function Sinh[z] in the complex plane
function Cosh[z] in the complex plane

Related pages

References

↑ ^1.0 ^1.1 Weisstein, Eric W.. Hyperbolic Functions (in en). mathworld.wolfram.com. Retrieved 2020-08-29.
↑ (1999) Collins Concise Dictionary, 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386
↑ ^3.0 ^3.1 Hyperbolic Functions. www.mathsisfun.com. Retrieved 2020-08-29.
↑ Woodhouse, N. M. J.. Special Relativity (2003). London: Springer. p. 71. ISBN 978-1-85233-426-0.
↑ Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972). New York: Dover Publications. ISBN 978-0-486-61272-0.

Cite error: <ref> tag with name "Osborn, 1902" defined in <references> is not used in prior text.

[:1-1] 1.0 ^1.1 Weisstein, Eric W.. Hyperbolic Functions (in en). mathworld.wolfram.com. Retrieved 2020-08-29.

[2] (1999) Collins Concise Dictionary, 4th edition, HarperCollins, Glasgow, ISBN 0 00 472257 4, p. 1386

[:2-3] 3.0 ^3.1 Hyperbolic Functions. www.mathsisfun.com. Retrieved 2020-08-29.

[4] Woodhouse, N. M. J.. Special Relativity (2003). London: Springer. p. 71. ISBN 978-1-85233-426-0.

[5] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1972). New York: Dover Publications. ISBN 978-0-486-61272-0.

[1]

[2]

[3]

[4]

[5]