اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFeynman integral in $mathbb R^1oplusmathbb R^m$ and complex expansion of $_2F_1$
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Closed form expressions are proposed for the Feynman integral $$ I_{D, m}(p,q) = intfrac{d^my}{(2pi)^m}intfrac{d^Dx}{(2pi)^D} frac1{(x-p/2)^2+(y-q/2)^4} frac1{(x+p/2)^2+(y+q/2)^4} $$ over $d=D+m$ dimensional space with $(x,y),,(p,q)in mathbb R^D oplus mathbb R^m$, in the special case $D=1$. We show that $I_{1,m}(p,q)$ can be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function $_2F_1$, as well as generalized hypergeometric $_2F_2$ and $_3F_2$, Horn $H_4$ and Appell $F_2$ functions. Several interesting relations are derived between the real and imaginary parts of $_2F_1$ and the function $H_4$.
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