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AsukaMinato
AsukaMinato
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Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]]r*Cos[ϕ],y -> r*Sin[\[Phi]]r*Sin[ϕ]}, {r, 0, R}]

(-1 + E^(R^2 (a Cos[\[Phi]]^2Cos[ϕ]^2 + b Cos[\[Phi]]Cos[ϕ] Sin[\[Phi]]Sin[ϕ] + c Sin[\[Phi]]^2Sin[ϕ]^2)))/(a + c + (a - c) Cos[2 \[Phi]]ϕ] + b Sin[2 \[Phi]]ϕ])

It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[\[Phi]ForAll[ϕ, \[Phi]ϕ > -Pi && \[Phi]ϕ <= Pi, a Cos[\[Phi]]^2Cos[ϕ]^2 + b Cos[\[Phi]]Cos[ϕ] Sin[\[Phi]]Sin[ϕ] + c Sin[\[Phi]]^2Sin[ϕ]^2 < 0] is valid as the result of

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]]r*Cos[ϕ], y -> r*Sin[\[Phi]]r*Sin[ϕ]}, {r, 0,Infinity},Assumptions -> {a, b, c} \[Element]∈ Reals && \[Phi]ϕ > -Pi && \[Phi]ϕ <= Pi]

ConditionalExpression[-(1/( 2 (a Cos[\[Phi]]^2Cos[ϕ]^2 + b Cos[\[Phi]]Cos[ϕ] Sin[\[Phi]]Sin[ϕ] + c Sin[\[Phi]]^2Sin[ϕ]^2))), a Cos[\[Phi]]^2Cos[ϕ]^2 + b Cos[\[Phi]]Cos[ϕ] Sin[\[Phi]]Sin[ϕ] + c Sin[\[Phi]]^2Sin[ϕ]^2 < 0]

Resolve[ForAll[\[Phi]Resolve[ForAll[ϕ, \[Phi]ϕ > -Pi && \[Phi]ϕ <= Pi, a Cos[\[Phi]]^2Cos[ϕ]^2 + b Cos[\[Phi]]Cos[ϕ] Sin[\[Phi]]Sin[ϕ] + c Sin[\[Phi]]^2Sin[ϕ]^2 < 0], Reals] 
Integrate[(-1)/(a + c + (a - c) Cos[2 \[Phi]]ϕ] + b Sin[2 \[Phi]]ϕ]), {\[Phi]ϕ, -Pi, Pi}, Assumptions -> c < 0 && a < b^2/(4 c) && a < 0 && b \[Element]∈ Reals]

, thinking a few hours. In fact, this is $$\frac{2 \pi }{\sqrt{4 a c-b^2}} $$ as Simplify under the conditions c < 0 && a < b^2/(4 c) && a < 0 && b \[Element]∈ Reals] shows.

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]],y -> r*Sin[\[Phi]]}, {r, 0, R}]

(-1 + E^(R^2 (a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2)))/(a + c + (a - c) Cos[2 \[Phi]] + b Sin[2 \[Phi]])

It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0] is valid as the result of

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]], y -> r*Sin[\[Phi]]}, {r, 0,Infinity},Assumptions -> {a, b, c} \[Element] Reals && \[Phi] > -Pi && \[Phi] <= Pi]

ConditionalExpression[-(1/( 2 (a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2))), a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0]

Resolve[ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0], Reals] 
Integrate[(-1)/(a + c + (a - c) Cos[2 \[Phi]] + b Sin[2 \[Phi]]), {\[Phi], -Pi, Pi}, Assumptions -> c < 0 && a < b^2/(4 c) && a < 0 && b \[Element] Reals]

, thinking a few hours. In fact, this is $$\frac{2 \pi }{\sqrt{4 a c-b^2}} $$ as Simplify under the conditions c < 0 && a < b^2/(4 c) && a < 0 && b \[Element] Reals] shows.

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[ϕ],y -> r*Sin[ϕ]}, {r, 0, R}]

(-1 + E^(R^2 (a Cos[ϕ]^2 + b Cos[ϕ] Sin[ϕ] + c Sin[ϕ]^2)))/(a + c + (a - c) Cos[2 ϕ] + b Sin[2 ϕ])

It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[ϕ, ϕ > -Pi && ϕ <= Pi, a Cos[ϕ]^2 + b Cos[ϕ] Sin[ϕ] + c Sin[ϕ]^2 < 0] is valid as the result of

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[ϕ], y -> r*Sin[ϕ]}, {r, 0,Infinity},Assumptions -> {a, b, c} ∈ Reals && ϕ > -Pi && ϕ <= Pi]

ConditionalExpression[-(1/( 2 (a Cos[ϕ]^2 + b Cos[ϕ] Sin[ϕ] + c Sin[ϕ]^2))), a Cos[ϕ]^2 + b Cos[ϕ] Sin[ϕ] + c Sin[ϕ]^2 < 0]

Resolve[ForAll[ϕ, ϕ > -Pi && ϕ <= Pi, a Cos[ϕ]^2 + b Cos[ϕ] Sin[ϕ] + c Sin[ϕ]^2 < 0], Reals] 
Integrate[(-1)/(a + c + (a - c) Cos[2 ϕ] + b Sin[2 ϕ]), {ϕ, -Pi, Pi}, Assumptions -> c < 0 && a < b^2/(4 c) && a < 0 && b ∈ Reals]

, thinking a few hours. In fact, this is $$\frac{2 \pi }{\sqrt{4 a c-b^2}} $$ as Simplify under the conditions c < 0 && a < b^2/(4 c) && a < 0 && b ∈ Reals] shows.

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user64494
user64494
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It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0] is valid as the result of

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]], y -> r*Sin[\[Phi]]}, {r, 0,Infinity},Assumptions -> {a, b, c} \[Element] Reals && \[Phi] > -Pi && \[Phi] <= Pi]

ConditionalExpression[-(1/( 2 (a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2))), a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0]

demonstrates. 

Unfortunately, Mathematica fails with

It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0] is valid. Unfortunately, Mathematica fails with

It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0] is valid as the result of

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]], y -> r*Sin[\[Phi]]}, {r, 0,Infinity},Assumptions -> {a, b, c} \[Element] Reals && \[Phi] > -Pi && \[Phi] <= Pi]

ConditionalExpression[-(1/( 2 (a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2))), a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0]

demonstrates. 

Unfortunately, Mathematica fails with

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user64494
user64494
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This can be done with help of Mathematica as follows. Let us exhaust the plane by disks centered at the origin. Switching to the polar coordinates, one obtains

Integrate[r*Exp[a*x^2 + b*x*y + c*y^2] /. {x -> r*Cos[\[Phi]],y -> r*Sin[\[Phi]]}, {r, 0, R}]

(-1 + E^(R^2 (a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2)))/(a + c + (a - c) Cos[2 \[Phi]] + b Sin[2 \[Phi]])

It is clear the finite limit of the function defined by the above expression as R->Infinity exists only if ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0] is valid. Unfortunately, Mathematica fails with

Resolve[ForAll[\[Phi], \[Phi] > -Pi && \[Phi] <= Pi, a Cos[\[Phi]]^2 + b Cos[\[Phi]] Sin[\[Phi]] + c Sin[\[Phi]]^2 < 0], Reals]

, returning the input. However, Mathematica is able to derive the condition in such a way

Resolve[ForAll[y, y >= -1 && y <= 1,a y^2 + b *y*Sqrt[1 - y^2] + c (1 - y^2) < 0], Reals]

c < 0 && a < b^2/(4 c)

Now

Integrate[(-1)/(a + c + (a - c) Cos[2 \[Phi]] + b Sin[2 \[Phi]]), {\[Phi], -Pi, Pi}, Assumptions -> c < 0 && a < b^2/(4 c) && a < 0 && b \[Element] Reals]

results in

ConditionalExpression[ -Piecewise[{{0, (Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) - Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] < 1 && Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) + Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] < 1) || (Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) - Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] >= 1 && Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) + Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] >= 1)}, {(2*Pi)/Sqrt[-b^2 + 4*a*c], Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) + Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] < 1}}, (-2*Pi)/Sqrt[-b^2 + 4*a*c]], !(Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) - Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] >= 1 && Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) - Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] <= 1) && !(Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) + Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] >= 1 && Sqrt[Abs[-(a/(a - I*b - c)) - c/(a - I*b - c) + Sqrt[-b^2 + 4*a*c]/(a - I*b - c)]] <= 1)]

, thinking a few hours. In fact, this is $$\frac{2 \pi }{\sqrt{4 a c-b^2}} $$ as Simplify under the conditions c < 0 && a < b^2/(4 c) && a < 0 && b \[Element] Reals] shows.