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TD n°5 : Résidus - Correction

 

Prolongement analytique

Exercice 1 :

Montrer que les séries suivantes sont des prolongements analytiques l’une de l’autre.

=

Calcul de résidus

Résidus.

Soit f est analytique partout à l’intérieur, et sur une courbe fermée simple 𝒞 excepté en    (qui es= t un pôle d’ordre n).

=

Exercice 2 :

Trouver les résidus des fonctions don= nées aux pôles précisés.

1°)

Le résidu de f en    est :

Le résidu de f en

Le résidu de f en    est :

2°)

Le résidu de g en    est : =

Le résidu de g en    est :   

3°)

Le résidu de h en    est :

 

 

4°)

Le résidu de k en    est :

 

 

 

Exercice 3 :

Trouver les résidus des fonctions don= nées aux pôles à distances finis.

1.&n= bsp;       définie par  .

2.&n= bsp;       définie par  .


 

Théorème des Résidus.

Théorème des Résidus.

Soit f est analytique partout à l’intérieur, et sur une courbe fermée simple 𝒞 excepté en un nombre fini de pôles= a, b, c … intérieurs à = 9966; alors :

Reprendre les fonctions de l’exercice= 2 et calculer :

1.&n= bsp;           pour   𝒞 définie par         =     

2.&n= bsp;           pour   𝒞 définie par           

3.&n= bsp;           pour   𝒞 définie par           

1°)

a)   Cfzdz=3D 2= 1-2i10+ 1+2i10=3D25

b)   Cfzdz=3D 2= 45+1-2i10+ 1+2i10=3D2<= span style=3D'font-size:11.0pt;line-height:115%;font-family:"Calibri","sans-seri= f"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:Calibri;mso-fareas= t-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times Ne= w Roman"; mso-bidi-theme-font:minor-bidi;position:relative;top:8.5pt;mso-text-raise:-= 8.5pt; mso-ansi-language:FR;mso-fareast-language:EN-US;mso-bidi-language:AR-SA'>

2°)

a)   Cgzdz=3D 2= 18=3D4

b)   Cgzdz=3D 2= 18-18=3D0

3°)

a)   Chzdz=3D 2 e3<= /i>t+3te3<= /i>t

 

 

 

 

 

Exercice 5 :

Calculer   pour   = 𝒞 définie par       

 

 

·         Le résidu au pôle simple <= !--[if gte msEquation 12]>z=3D1  est :

·         Le résidu au pôle double    est :

 

a)  z=3D32 donc      <= /span>Cfzdz=3D 2= e16=3Diπe<= /m:r>8

b)  z=3D10 donc=    Cfzdz=3D 2= e16+-5e-316=3D e= -= 5<= /i>e-3= 8

 

 

 

 

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