Markiyan Hirnyk

## 7228 Reputation

11 years, 334 days

## Precedent...

See https://www.mapleprimes.com/questions/200112-Line-Intersecting-A-Plane . Also all that is described in ?geom3d:-intersection.

## Explanation...

@Kitonum Maple produces a generic result which does not work for integer values of m. If m=-3, the integral under consideration diverges. Don't hesitate to ask for further explanation in need.

## Improvement...

```restart; value(int((x-a)^m/x, x = a .. x)) assuming x > a, a > 0, m>-1;
(hypergeom([-m, -m], [-m+1], a/x)*x^m-Pi*csc(Pi*m)*m*a^m)/m```

## Strong assumption...

`m::posint`

Do you find it reasonable?

## Mma says...

`int((x-a)^m/x, x) = (1-a/x)^(-m)*(-a+x)^m*hypergeom([-m, -m], [1-m], a/x)/m`

## Not verified...

• ```pdetest(u(x, y) = U2, PDE);

Error, (in expand/bigprod) Maple was unable to allocate enough memory to complete this computation.
```

PS. Also see the result of

```eval(U2, y = 1/x);
`assuming`([simplify(%)], [x > 0, x < 1]);
plot3d(U2, x = 0 .. 1, y = 0 .. 3, grid = [100, 100])```

## Indeed...

`plots:-complexplot3d(MeijerG([[1/2], []], [[], [1]], z), z = -2-2*I .. 2+2*I, axes = frame);`

Mma confirms it.

## Is the result a solution at all?...

@Thomas Richard Maple produces the space curve

```{u(_s) = _C1+_C2*sin(sqrt(3)*_s)+_C3*cos(sqrt(3)*_s),
x(_s) = -(1/2)*_C2*sin(sqrt(3)*_s)-(1/2)*_C3*cos(sqrt(3)*_s)+
_C1+(1/2)*_C2*sqrt(3)*cos(sqrt(3)*_s)-(1/2)*_C3*sqrt(3)*sin(sqrt(3)*_s),
y(_s) = -(1/2)*_C2*sqrt(3)*cos(sqrt(3)*_s)+(1/2)*_C3*sqrt(3)*sin(sqrt(3)*_s)-(1/2)*_C2*sin(sqrt(3)*_s)-
(1/2)*_C3*cos(sqrt(3)*_s)+_C1}```

The question arises: do there exist the partial dervatives of u(x,y) by x and y?

## Explanation...

@acer You wrote

• whatever "direct" means

I have had in mind the parametric option.

## Tricky way...

@acer Mma directly does it through

```Solve[{x^2 + y^2 + z^2 == 3, x + y + z == 3}, {x, y, z}, Reals]
{{x -> 1, y -> 1, z -> 1}}```

## Maple correctly answers a correctly pose...

Up to Maple help, the output of

```solve(sqrt(x) = -I, [x]);
[]
```

means the equation has no solution.

## Can't reproduce it...

I obtain

```restart; coulditbe(2+3*I < 0);
false
interface(version);
Standard Worksheet Interface, Maple 2017.3, Windows 10, September 27 2017 Build ID 1265877
```

coulditbe.mw

## Explanation...

@Kitonum In the case n::posint the set

` {simplify(evalc(solve(x^n = a+I*b, x, AllSolutions)))}`

is finite.

```restart; sol := simplify(evalc(solve(x^n = a+I*b, x, AllSolutions)));
sol := (a^2+b^2)^(1/(2*n))*(I*sin((arctan(b, a)+2*Pi*_Z1)/n)+cos((arctan(b, a)+2*Pi*_Z1)/n))
Originally _Z1, renamed _Z1~:
is assumed to be: integer
```

## Simplifying it...

```e2_3 := w^(1-sigma)*((f__11*sigma*Delta__1/L__1)^(-1/(sigma-1))/w)^(k-sigma+1)*k/
(Delta__1*(k-sigma+1)*a__0^k);
simplify(%, symbolic);
w^(-k)*f__11^((-k+sigma-1)/(sigma-1))*sigma^((-k+sigma-1)/(sigma-1))*
Delta__1^(-k/(sigma-1))*L__1^((k-sigma+1)/(sigma-1))*k*a__0^(-k)/(k-sigma+1)```

simplifying.mw

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