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The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. The converse does hold e.g. if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is ...
In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved.
The theory was completely resolved by Moeglin–Waldspurger, and was reverse-engineered to establish the "converse theorem". Symmetric square on GL(n) due to Shimura, and Gelbart–Jacquet (n = 2), Piatetski-Shapiro and Patterson (n = 3), and Bump–Ginzburg (n > 3). Exterior square on GL(n), due to Jacquet–Shalika and Bump–Ginzburg.
The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool. Since f is the pointwise limit of the sequence ( f n ) of measurable functions that are dominated by g , it is also measurable and dominated by g , hence it is integrable.
Dominated convergence theorem (Lebesgue integration) Donaldson's theorem (differential topology) Donsker's theorem (probability theory) Doob decomposition theorem (stochastic processes) Doob's martingale convergence theorems (stochastic processes) Doob–Meyer decomposition theorem (stochastic processes) Dudley's theorem (probability)
In 1936, André Weil proved a converse (of sorts) to Haar's theorem, by showing that if a group has a left invariant measure with a certain separating property, [3] then one can define a topology on the group, and the completion of the group is locally compact and the given measure is essentially the same as the Haar measure on this completion.