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<Input>
<Text-field style="Title" family="Arial" size="24" layout="Title"><Font family="Arial" size="24">Inverse of the Error Function</Font></Text-field>
</Input>
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<Group labelreference="L45" drawlabel="true">
<Input>
<Text-field style="Author" layout="Author">Univ.-Prof. Dr.-Ing. habil. J. BETTEN
RWTH University Aachen
Mathematical Models in Materials Science and Continuum Mechanics
Augustinerbach  4 - 22
D-52064  A a c h e n   /  Germany
<Hyperlink linktarget="mailto:betten@mmw.rwth-aachen.de" hyperlink="true"><Font style="Author" underline="true" foreground="[0,128,128]">betten@mmw.rwth-aachen.de</Font></Hyperlink></Text-field>
<Text-field style="Author" layout="Author"></Text-field>
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<Group labelreference="L47" drawlabel="true">
<Input>
<Text-field style="Text" bold="true" layout="Normal"><Font bold="true">Abstract</Font></Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">Forming the inverse of  the <Font italic="true" style="Text">GAUSS </Font>error function erf(x) a best approximation to erf(x) has</Text-field>
<Text-field style="Normal" layout="Normal">been discussed, the inverse of which can easily be determined.</Text-field>
<Text-field style="Normal" layout="Normal">   The deviation between the error function and its best approximation has been calculated</Text-field>
<Text-field style="Normal" layout="Normal">by considering the L-two error norm.</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal"><Font italic="true" style="Text">Keywords:   </Font>Best approximation; approximant; inverse; L-two error norm</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Text" bold="true" layout="Normal"></Text-field>
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<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field></Title>
<Group labelreference="L49" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">As has been explained by <Font italic="true" style="Text">BETTEN </Font>(2005) in more detail the creep behavior of several materials</Text-field>
<Text-field style="Normal" layout="Normal">can be interpreted as a <Font italic="true" style="Text">diffusion controlled process.</Font> Calculating this process we need the inverse</Text-field>
<Text-field style="Normal" layout="Normal">of the error function.</Text-field>
<Text-field style="Normal" layout="Normal">   The <Font italic="true" style="Text">GAUSS error function </Font>is implied in the Maple software as a standard function.</Text-field>
<Text-field style="Normal" layout="Normal">An approximation of its inverse is discussed in the following.</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
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<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Best Approximation</Text-field></Title>
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<Input>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal">An approximation to the error function erf(xi) on [0,r] is assumed by the <Font italic="true" style="Text">hyperbolic tangent:</Font></Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">restart:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">approximant:=tanh(a*xi);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JSxhcHByb3hpbWFudEctJSV0YW5oRzYjKiYlImFHIiIiJSN4aUciIiI=</Equation></Text-field>
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<Input>
<Text-field style="Normal" layout="Normal">This function is suitable because it is similar to  erf(xi). Furthermore, the inverse can easily be determined:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">inverse:=(1/a)*arctanh(xi);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JShpbnZlcnNlRyomJSJhRyEiIi0lKGFyY3RhbmhHNiMlI3hpRyIiIg==</Equation></Text-field>
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<Text-field style="Normal" layout="Normal">Note that the <Font italic="true" style="Text">Area Tangent, </Font>artanh(...), is indicated as arctanh(...) in MAPLE.  The <Font italic="true" style="Text">best approximation </Font>by the <Font italic="true" style="Text">hyperbolic tangent </Font>is guaranteed by an  <Font italic="true" style="Text">optimal parameter</Font> a  which minimizes the  L-two error norm:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L_two[0..r]:=sqrt((1/r)*int((erf(xi)-tanh(a*xi))^2, xi=0..r))=minimum;</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhJSJyRy8qJCksJComKiQlInJHISIiIiIiLSUkaW50RzYkKiQpLCYtJSRlcmZHNiMlI3hpRyIiIi0lJXRhbmhHNiMqJiUiYUciIiIlI3hpRyIiIiEiIiIiIyIiIi8lI3hpRzsiIiElInJHIiIiIiIiIyIiIiIiIyIiIiUobWluaW11bUc=</Equation></Text-field>
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<Text-field style="Normal" layout="Normal">Thus, the derivative of the integral should be equal to zero.</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">derivative[0..r]:=diff(int((erf(xi)-tanh(a*xi))^2, xi=0..r),a);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUrZGVyaXZhdGl2ZUc2IzsiIiElInJHLSUkaW50RzYkLCQqKiIiIyIiIiwmLSUkZXJmRzYjJSN4aUciIiItJSV0YW5oRzYjKiYlImFHIiIiJSN4aUciIiIhIiIiIiIsJiIiIiIiIiokKS0lJXRhbmhHNiMqJiUiYUciIiIlI3hpRyIiIiIiIyIiIiEiIiIiIiUjeGlHIiIiISIiLyUjeGlHOyIiISUickc=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L7" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">Depending on the range  [0, r] considered, we obtain  the following optimal parameters:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for r in [1,2,3,4,5,infinity] do 
a[optimal][0..r]:= fsolve(int(xi*(erf(xi)-tanh(a_*xi))*(1-(tanh(a_*xi))^2), xi=0..r)=0,a_) 
od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiYlImFHNiMlKG9wdGltYWxHNiM7IiIhIiIiJCIrOyRvRzwiISIq</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiYlImFHNiMlKG9wdGltYWxHNiM7IiIhIiIjJCIrTjRGLDchIio=</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiYlImFHNiMlKG9wdGltYWxHNiM7IiIhIiIkJCIrIWVnRj8iISIq</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiYlImFHNiMlKG9wdGltYWxHNiM7IiIhIiIlJCIrIkcjeS03ISIq</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiYlImFHNiMlKG9wdGltYWxHNiM7IiIhIiImJCIrOkR5LTchIio=</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiYlImFHNiMlKG9wdGltYWxHNiM7IiIhJSlpbmZpbml0eUckIis8RHktNyEiKg==</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L8" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">The corresponding  L-two error norms are given as:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for r in [1,2,3,4,5,infinity] do 
L_two[0..r]:=evalf(sqrt((1/r)*int((erf(xi)-tanh(a[optimal][0..r]*xi))^2,xi=0..r))) 
od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhIiIiJCIrZVMmKj4jKSEjNw==</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhIiIjJCIrSSFRKUc5ISM2</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhIiIkJCIrdTgwOzchIzY=</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhIiIlJCIrISkpXE8wIiEjNg==</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhIiImJCIrLSVwVFUqISM3</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhJSlpbmZpbml0eUckIiIhIiIh</Equation></Text-field>
</Output>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Inverse Functions</Text-field></Title>
<Group labelreference="L11" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">Depending on the range  [0, r]  we obtain the following inverse functions:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for r in [1,2,3,4,5,infinity] do 
inverse[0..r]:=(1/a[optimal][0..r])*arctanh(xi) 
od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUoaW52ZXJzZUc2IzsiIiEiIiIsJComJCIrYGo1RSYpISM1IiIiLSUoYXJjdGFuaEc2IyUjeGlHIiIiIiIi</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUoaW52ZXJzZUc2IzsiIiEiIiMsJComJCIrTW5eQyQpISM1IiIiLSUoYXJjdGFuaEc2IyUjeGlHIiIiIiIi</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUoaW52ZXJzZUc2IzsiIiEiIiQsJComJCIrU20/OSQpISM1IiIiLSUoYXJjdGFuaEc2IyUjeGlHIiIiIiIi</Equation></Text-field>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUoaW52ZXJzZUc2IzsiIiEiIiUsJComJCIrS20wOSQpISM1IiIiLSUoYXJjdGFuaEc2IyUjeGlHIiIiIiIi</Equation></Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUoaW52ZXJzZUc2IzsiIiElKWluZmluaXR5RywkKiYkIissXTA5JCkhIzUiIiItJShhcmN0YW5oRzYjJSN4aUciIiIiIiI=</Equation></Text-field>
</Output>
</Group>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Examples</Text-field></Title>
<Group labelreference="L12" drawlabel="true">
<Input>
<Text-field style="Text" bold="true" layout="Normal"></Text-field>
</Input>
</Group>
<Group labelreference="L13" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">In the following some examples should be plotted:</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">alias(H=Heaviside,th=thickness):</Text-field>
</Input>
</Group>
<Group labelreference="L14" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot({1,H(xi-2),erf(xi),tanh(a[optimal][0..2]*xi)}, xi=0..2.001,color=black);</Text-field>
</Input>
<Output>
<Text-field style="Maple Plot" layout="Maple Plot"><Plot height="224" type="two-dimensional" width="309" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" 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</Output>
</Group>
<Group labelreference="L15" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">Delta(xi):=erf(xi)-tanh(a[optimal][0..2]*xi);</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+LSUmRGVsdGFHNiMlI3hpRywmLSUkZXJmRzYjJSN4aUciIiItJSV0YW5oRzYjLCQqJiQiK040Riw3ISIqIiIiJSN4aUciIiIiIiIhIiI=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L16" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i from 1 to 3 do 
zero[i-1]:=fsolve(Delta(xi)=0,xi,(i-1)/2..i/2) 
od;</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUlemVyb0c2IyIiISQiIiEiIiE=</Equation></Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUlemVyb0c2IyIiIy0lJ2Zzb2x2ZUc2JS8sJi0lJGVyZkc2IyUjeGlHIiIiLSUldGFuaEc2IywkKiYkIitONEYsNyEiKiIiIiUjeGlHIiIiIiIiISIiIiIhJSN4aUc7IiIiIyIiJCIiIw==</Equation></Text-field>
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<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot1:=plot({-0.02,0.02,0.02*H(xi-2),-0.02*H(xi-2)}, xi=0..2.001,color=black):</Text-field>
</Input>
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<Group labelreference="L18" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot2:=plot(Delta(xi),xi=0..2,color=black,th=3, title=&quot;deviation between erf(xi) and tanh(a*xi) on [0, 2]&quot;):</Text-field>
</Input>
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<Group labelreference="L19" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plots[display]({plot1,plot2});</Text-field>
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<Output>
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<Group labelreference="L20" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">L_two[0..2]:=sqrt((1/2)*Int((Delta)^2,xi=0..2))= evalf(sqrt((1/2)*int((Delta(xi))^2,xi=0..2)));</Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmTF90d29HNiM7IiIhIiIjLywkKiYjIiIiIiIjIiIiKiYpIiIjIyIiIiIiIyIiIiktJSRJbnRHNiQqJCklJkRlbHRhRyIiIyIiIi8lI3hpRzsiIiEiIiMjIiIiIiIjIiIiIiIiIiIiJCIrSSFRKUc5ISM2</Equation></Text-field>
</Output>
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<Group labelreference="L21" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">for i in [1.18,1.19,a[optimal][0..2],1.21,1.22] do 
L_two[i]:=evalf(sqrt((1/2)*int((erf(xi)-tanh(i*xi))^2, xi=0..2))) 
od;</Text-field>
</Input>
<Output>
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<Group labelreference="L22" drawlabel="true">
<Input>
<Text-field style="Normal" layout="Normal">For the value  a =1.201270935  the L-two error norm is minimal in the range  xi = [0, 2].  The following Figures illustrate the error function  erf(xi)  and some inverse  approximations  (1/a)*arctanh(xi):</Text-field>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot1:=plot({1,xi,H(xi-1)}, xi=0..1.001,scaling=constrained,color=black):</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot2:=plot({erf(xi),(1/a[optimal][0..1])*arctanh(xi)}, xi=0..1,0..1,scaling=constrained,color=black,th=2):</Text-field>
</Input>
</Group>
<Group labelreference="L24" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plots[display]({plot1,plot2});</Text-field>
</Input>
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<Text-field style="Normal" layout="Normal">   The results show that the approximant  <Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEldGFuaEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSSNtb0dGJDYtUScmc2RvdDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUSN4aUYnRi9GMkYyRjJGMg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEldGFuaEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSSNtb0dGJDYtUScmc2RvdDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUSN4aUYnRi9GMkYyRjJGMg==</Equation> furnishes a suitable approximation to the error function <Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkZXJmRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRI3hpRidGL0YyRjJGMkYy">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkZXJmRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRI3hpRidGL0YyRjJGMkYy</Equation>. In view of the inverse <Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZKLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjdGNy1GPTYlUShhcmN0YW5oRicvRkFGT0Y3LUYsNiQtRiM2JC1GPTYlUSN4aUYnRlNGN0Y3RjdGNw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y4USdpdGFsaWNGJ0Y3LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZKLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjdGNy1GPTYlUShhcmN0YW5oRicvRkFGT0Y3LUYsNiQtRiM2JC1GPTYlUSN4aUYnRlNGN0Y3RjdGNw==</Equation> and the approximant <Equation executable="false" style="Normal" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEldGFuaEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSSNtb0dGJDYtUScmc2RvdDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUSN4aUYnRi9GMkYyRjJGMg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEldGFuaEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJhRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSSNtb0dGJDYtUScmc2RvdDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1GLDYlUSN4aUYnRi9GMkYyRjJGMg==</Equation> the influence of  the range [0, r] on the parameter  a  is less important.</Text-field>
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<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output">NiM+JiUmZGVsdGFHNiQlKWluZmluaXR5RyIiIy8sJiYtJSV0YW5oRzYjKiYlImFHIiIiJSN4aUciIiI2IyUpaW5maW5pdHlHIiIiJi0lJXRhbmhHNiMqJiUiYUciIiIlI3hpRyIiIjYjIiIjISIiLCYtJSV0YW5oRzYjLCQqJiQiKzxEeS03ISIqIiIiJSN4aUciIiIiIiIiIiItJSV0YW5oRzYjLCQqJiQiK040Riw3ISIqIiIiJSN4aUciIiIiIiIhIiI=</Equation></Text-field>
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<Group labelreference="L34" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot1:=plot({H(xi-5),tanh(1.202782517*xi),tanh(1.172868316*xi)}, xi=0..5.001,color=black):</Text-field>
</Input>
</Group>
<Group labelreference="L35" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot2:=plot(1,xi=0..5,color=black, title=&quot;two approximants tanh(a*xi)&quot;):</Text-field>
</Input>
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<Group labelreference="L36" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plots[display]({plot1,plot2});</Text-field>
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<Output>
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<Group labelreference="L37" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot1:= plot({0.012*H(xi-2),tanh(1.202782517*xi)-tanh(1.172868316*xi), tanh(1.202782517*xi)-tanh(1.201270935*xi)}, xi=0..2.001,color=black):</Text-field>
</Input>
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<Group labelreference="L38" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plot2:=plot(0.012,xi=0..2,color=black, title=&quot;distance delta [..] between approximants&quot;):</Text-field>
</Input>
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<Group labelreference="L39" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal">plots[display]({plot1,plot2});</Text-field>
</Input>
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<Text-field style="Normal" layout="Normal">   The approximation dicussed above has been published in:</Text-field>
<Text-field style="Normal" layout="Normal"><Font italic="true" style="Text">BETTEN, J.: </Font><Font bold="true" style="Text">Creep Mechanics, </Font>Springer-Verlag, Berlin/Heidelberg/New York, 2nd Edition 2005.</Text-field>
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