Calc1.12

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\int e^{{x^{2}}}\,dx\, and \int _{0}^{L}e^{{x^{2}}}\,dx\,

The integral at all limit combinations involving infinity equals infinity. From Maple,

\int e^{{x^{2}}}\,dx={-1 \over 2}i{\sqrt  {\pi }}\operatorname {erf}(xi)\,

\int _{0}^{L}e^{{x^{2}}}\,dx={-1 \over 2}i{\sqrt  {\pi }}\operatorname {erf}(Li)\,


Where erf is the error function and is defined as

\operatorname {erf}(x)={\frac  {2}{{\sqrt  {\pi }}}}\int _{0}^{x}e^{{-t^{2}}}dt={\frac  {2}{{\sqrt  {\pi }}}}\sum _{{n=0}}^{\infty }{\frac  {(-1)^{n}x^{{2n+1}}}{(2n+1)n!}}={\frac  {2}{{\sqrt  {\pi }}}}\left(x-{\frac  {x^{3}}{3}}+{\frac  {x^{5}}{10}}-{\frac  {x^{7}}{42}}+{\frac  {x^{9}}{216}}-\ \cdots \right)

={\frac  {2x}{{\sqrt  {\pi }}}}{}_{1}F_{1}\left({\frac  {1}{2}};{\frac  {3}{2}};-x^{2}\right)\,

For example, according to Maple,

\operatorname {erf}(i)=1.650425759i\,

\operatorname {erf}(10i)=.1524307423\cdot 10^{{43}}i\,


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