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Tanh






Mathematica Notation

Traditional Notation









Elementary Functions > Tanh[z] > Transformations > Addition formulas





http://functions.wolfram.com/01.21.16.0035.01









  


  










Input Form





Tanh[Subscript[z, 1] + Subscript[z, 2] + Subscript[z, 3]] == (Tanh[Subscript[z, 1]] + Tanh[Subscript[z, 2]] + Tanh[Subscript[z, 3]] + Tanh[Subscript[z, 1]] Tanh[Subscript[z, 2]] Tanh[Subscript[z, 3]])/ (1 + Tanh[Subscript[z, 1]] Tanh[Subscript[z, 2]] + Tanh[Subscript[z, 1]] Tanh[Subscript[z, 3]] + Tanh[Subscript[z, 2]] Tanh[Subscript[z, 3]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Tanh", "[", RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"], "+", SubscriptBox["z", "3"]]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["z", "1"], "]"]], "+", RowBox[List["Tanh", "[", SubscriptBox["z", "2"], "]"]], "+", RowBox[List["Tanh", "[", SubscriptBox["z", "3"], "]"]], "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["z", "1"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["z", "2"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["z", "3"], "]"]]]]]], RowBox[List["1", "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["z", "1"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["z", "2"], "]"]]]], "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["z", "1"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["z", "3"], "]"]]]], "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["z", "2"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["z", "3"], "]"]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> tanh </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <tanh /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <tanh /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Tanh", "[", RowBox[List[SubscriptBox["z_", "1"], "+", SubscriptBox["z_", "2"], "+", SubscriptBox["z_", "3"]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["zz", "1"], "]"]], "+", RowBox[List["Tanh", "[", SubscriptBox["zz", "2"], "]"]], "+", RowBox[List["Tanh", "[", SubscriptBox["zz", "3"], "]"]], "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["zz", "2"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["zz", "3"], "]"]]]]]], RowBox[List["1", "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["zz", "2"], "]"]]]], "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["zz", "3"], "]"]]]], "+", RowBox[List[RowBox[List["Tanh", "[", SubscriptBox["zz", "2"], "]"]], " ", RowBox[List["Tanh", "[", SubscriptBox["zz", "3"], "]"]]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





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