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The remainder theorem in Analyze complexes is a powerful tool to evaluate curvilinear integral of holomorphic functions on closed curves; it can as well be used to calculate integrals of real functions as the sum of certain series. It generalizes the integral Théorème of Cauchy and the integral Formule of Cauchy.

The proposal is the following one. Let us suppose that U is an open subset and Simplement related of the plane complex $\ mathbb C$ , Z ₁,…, Z _N a finished number of points distinct from U and F is a function which is definite and holomorphic on U - { Z ₁,…, Z _N}. If γ is a rectifiable Courbe in U which does not meet any the singular points Z _K and whose starting point is equal to the point of arrival (i.e. a rectifiable lace), then:

$\ oint_ \ gamma F (Z) \ text {D} Z =$

2 \ pi I \ sum_ {k=1} ^n \ operatorname {I} (\ gamma, z_k) \, \ operatorname {LMBO} (F, z_k)

Here, LMBO ( F , Z _K) indicates the residue F in Z _K, and I (γ, Z _K) the index of the lace γ of in $\ mathbb {C}$ compared to Z _K. Intuitively, it is the number of revolutions around Z _K carried out by a point describing all the lace. This number of revolutions is a whole ; it is positive if γ is traversed in the opposite direction of the needles of a watch around Z _K, no one if γ does not move at all around Z _K, and negative if γ is traversed in the direction of the needles of a watch around Z _K.

The index is defined by: $\ operatorname {I} (\ gamma, z_k) = \ frac {1} {2 \ pi I} \ int_ {\ gamma} \ frac {\ text {D} Z} {z-z_k}$

If $\ gamma$ is a Courbe of Jordan, then I (γ, Z _K) = $\ pm$ 1: the plus sign (+) if the curve is traversed in the direct direction (trigonometrical direction), the minus sign (-) if the curve is traversed in the indirect direction (direction of the needles of a watch). In this case, one a:

$\ oint_ \ gamma F (Z) \ text {D} Z =$

\ pm 2 \ pi I \ sum_ {k=1} ^n \ operatorname {LMBO} (F, z_k)

To evaluate real integral , the remainder theorem is used often in the following way: the intégrande is prolonged in a holomorphic Fonction on open of the complex plan; its Résidu S is calculated (what is usually easy), and part of the real axis is extended to a curve closed by attaching a half-circle in the higher or lower half-plane to him. The integral according to this curve can then be calculated by using the remainder theorem. Often, the part of the integral on the half-circle tends towards zero (Lemme of Jordan), when the ray of this last tends towards the infinite one, leaving only the part of the integral on the real axis, that which initially interested us.

This theorem makes it possible to simply calculate many integrals of real functions of real variables, for example:

: $\ int_ {0} ^ {+ \ infty} \ frac {\ sin x^2} {X} \ text {D} x= \ frac {\ pi} {4}$

: $\ int_ {0} ^ {+ \ infty} \ frac {\ ln X} {x^2-1} \ text {D} x= \ frac {\ pi^2} {4}$

This theorem is used in a crucial way in the analytical demonstrations of the Théorème of the prime numbers.

Types of integrals

One counts five usual types of integrals which one can easily calculate the value using the remainder theorem:

: 1st type: $\ int_ {0} ^ {2 \ pi} R (\ cos T, \ sin T) \ text {D} T \$ , where R indicates a rational fraction without pole on the unit $\ {x^2 + y^2 = 1 \}$ .

By posing $F (Z) = \ frac {1} {iz} R \ left (\ frac {Z + \ frac {1} {Z}} {2}, \ frac {Z - \ frac {1} {Z}} {2i} \ right)$ , the remainder theorem gives: $\ int_ {0} ^ {2 \ pi} R (\ cos T, \ sin T) \ text {D} T = 2i \ pi \ sum_ {W \ text {pole of} F \ text {such as} |W| < 1} \ operatorname {LMBO} (F, W)$

: 2nd type: $\ int_ {- \ infty} ^ {+ \ infty} R (X) \ text {D} X \$ , where R indicates a rational fraction without real pole, such as $\ lim_ {X \ rightarrow + \ infty} X R (X) = 0$ .

The remainder theorem gives: $\ int_ {- \ infty} ^ {+ \ infty} R (X) \ text {D} X \ = 2i \ pi \ sum_ {W, \ text {pole of R Im} (W) >0} \ operatorname {LMBO} (R, W)$

: 3rd type: $\ int_ {- \ infty} ^ {+ \ infty} F (X) e^ {ix} \ text {D} X \$ , where F is a holomorphic fraction in the vicinity of any point of the half-plane $\ {there \ geqslant 0 \}$ .

The remainder theorem gives that if $\ lim_$

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