The RiemannPsols routine returns either a list of solutions of a second order linear differential equation having three regular singular points or an equation of which the solutions have been multiplied by an exponential integral.

There are two input forms. The first has as the first argument a linear differential equation in diff or D form and as the second argument the variable in the differential equation.

A second input sequence accepts for the first argument a list of coefficients of the linear ode, and for the second argument the independent variable of the lode. This input sequence is useful for programming with the RiemannPsols routine.

In the second calling sequence, the list of coefficients is given in order from low differential order to high differential order and does not include the nonhomogeneous term.

This function is part of the DEtools package, and so it can be used in the form RiemannPsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[RiemannPsols](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{ode}≔x\left(1-x\right)\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\left(c-\left(a+b+1\right)x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-aby\left(x\right)$

$\mathrm{ode}≔x\left(1-x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\left(c-\left(a+b+1\right)x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-aby\left(x\right)$

$\mathrm{RiemannPsols}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[\mathrm{hypergeom}\left(\left[a\,b\right]\,\left[c\right]\,x\right)\,x^{-c+1}\mathrm{hypergeom}\left(\left[1+a-c\,1+b-c\right]\,\left[2-c\right]\,x\right)\right]$

$A≔\left[-ab\,c-\left(a+b+1\right)x\,x\left(1-x\right)\right]$

$A≔\left[-ab\,c-\left(a+b+1\right)x\,x\left(1-x\right)\right]$

$\mathrm{RiemannPsols}\left(A\,x\right)$

$\left[\mathrm{hypergeom}\left(\left[a\,b\right]\,\left[c\right]\,x\right)\,x^{-c+1}\mathrm{hypergeom}\left(\left[1+b-c\,1+a-c\right]\,\left[2-c\right]\,x\right)\right]$

$\mathrm{ode}≔\left(1-x^2\right)\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-2x\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(v\left(v+1\right)-\frac{u^2}{1-x^2}\right)y\left(x\right)$

$\mathrm{ode}≔\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(v\left(v+1\right)-\frac{u^2}{-x^2+1}\right)y\left(x\right)$

$\mathrm{RiemannPsols}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[\mathrm{hypergeom}\left(\left[-v+u\,u+v+1\right]\,\left[1+u\right]\,\frac{x}{2}+\frac{1}{2}\right){\left(x+1\right)}^{\frac{u}{2}}{\left(x-1\right)}^{\frac{u}{2}}\,{\left(\frac{x}{2}+\frac{1}{2}\right)}^{-u}\mathrm{hypergeom}\left(\left[v+1\,-v\right]\,\left[1-u\right]\,\frac{x}{2}+\frac{1}{2}\right){\left(x+1\right)}^{\frac{u}{2}}{\left(x-1\right)}^{\frac{u}{2}}\right]$