Difference between revisions of "Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z)"
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(Created page with "==Theorem== The following formula holds for $\mathrm{Re}(z)>0$: $$2e^{z^2}\displaystyle\int_z^{\infty} e^{-t^2} \mathrm{d}t = \dfrac{1}{z+\dfrac{\frac{1}{2}}{z+\dfrac{1}{z+\df...") |
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==Theorem== | ==Theorem== | ||
The following formula holds for $\mathrm{Re}(z)>0$: | The following formula holds for $\mathrm{Re}(z)>0$: | ||
| − | $$2e^{z^2}\displaystyle\int_z^{\infty} e^{-t^2} \mathrm{d}t = \dfrac{1}{z+\dfrac{\frac{1}{2}}{z+\dfrac{1}{z+\dfrac{\frac{3}{2}}{z+\dfrac{2}{z+\ldots}}}}}$$ | + | $$2e^{z^2}\displaystyle\int_z^{\infty} e^{-t^2} \mathrm{d}t = \dfrac{1}{z+\dfrac{\frac{1}{2}}{z+\dfrac{1}{z+\dfrac{\frac{3}{2}}{z+\dfrac{2}{z+\ldots}}}}},$$ |
| + | where $e^{z^2}$ denotes the [[exponential]] and the right hand side denotes a [[continued fraction]]. | ||
==Proof== | ==Proof== | ||
==Refrences== | ==Refrences== | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x|next=Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt}}: 7.1.14 | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Two-sided inequality for e^(x^2) integral from x to infinity e^(-t^2) dt for non-negative real x|next=Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt}}: 7.1.14 | ||
| + | |||
| + | [[Category:Theorem]] | ||
Latest revision as of 02:08, 6 June 2016
Theorem
The following formula holds for $\mathrm{Re}(z)>0$: $$2e^{z^2}\displaystyle\int_z^{\infty} e^{-t^2} \mathrm{d}t = \dfrac{1}{z+\dfrac{\frac{1}{2}}{z+\dfrac{1}{z+\dfrac{\frac{3}{2}}{z+\dfrac{2}{z+\ldots}}}}},$$ where $e^{z^2}$ denotes the exponential and the right hand side denotes a continued fraction.
Proof
Refrences
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 7.1.14
