Quiz — dl-11-12-FctsHolom.pdf

Question 1 sur 10 20:00

[{"id":21599,"question":"Quelle condition doit vérifier une fonction complexe pour être holomorphe ?","option_a":"Être continue","option_b":"Être dérivable en tout point","option_c":"Être bornée","option_d":"Être périodique","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction holomorphe doit être dérivable en tout point de son domaine de définition.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Être continue\", \"b\": \"Être dérivable en tout point\", \"c\": \"Êt","_debug_options_count":4},{"id":21600,"question":"Les équations de Cauchy-Riemann relient :","option_a":"La partie réelle et la partie imaginaire","option_b":"La dérivée et l'intégrale","option_c":"Le module et l'argument","option_d":"La continuité et la dérivabilité","option_e":"","option_f":"","bonne_reponse":"a","explication":"Elles relient les dérivées partielles de la partie réelle et imaginaire d'une fonction complexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"La partie réelle et la partie imaginaire\", \"b\": \"La dérivée et","_debug_options_count":4},{"id":21601,"question":"Le théorème de Cauchy s'applique à :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème de Cauchy s'applique aux fonctions holomorphes sur un domaine simplement connexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":21602,"question":"Quelle est la dérivée de la fonction f(z) = z² ?","option_a":"2z","option_b":"z","option_c":"z²","option_d":"2z²","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de z² est 2z, comme en analyse réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2z\", \"b\": \"z\", \"c\": \"z²\", \"d\": \"2z²\"}}","_debug_options_count":4},{"id":21603,"question":"Une fonction holomorphe est toujours :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction holomorphe est dérivable, mais pas nécessairement continue (cas des singularités).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":21604,"question":"Quelle propriété caractérise une fonction holomorphe ?","option_a":"Elle est bijective","option_b":"Elle vérifie les équations de Cauchy-Riemann","option_c":"Elle est périodique","option_d":"Elle est constante","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les équations de Cauchy-Riemann sont une condition nécessaire et suffisante pour la holomorphie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Elle est bijective\", \"b\": \"Elle vérifie les équations de Cauchy","_debug_options_count":4},{"id":21605,"question":"Le théorème des résidus permet de calculer :","option_a":"Des intégrales réelles","option_b":"Des intégrales complexes","option_c":"Des dérivées","option_d":"Des limites","option_e":"","option_f":"","bonne_reponse":"b","explication":"Il permet de calculer des intégrales complexes en utilisant les pôles de la fonction.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Des intégrales réelles\", \"b\": \"Des intégrales complexes\", \"c\":","_debug_options_count":4},{"id":21606,"question":"Une fonction holomorphe sur tout le plan complexe est dite :","option_a":"Entière","option_b":"Partielle","option_c":"Singulière","option_d":"Bornée","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction holomorphe sur tout le plan complexe est appelée fonction entière.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Entière\", \"b\": \"Partielle\", \"c\": \"Singulière\", \"d\": \"Bornée\"}}","_debug_options_count":4},{"id":21607,"question":"Quelle est la dérivée de la fonction f(z) = e^z ?","option_a":"e^z","option_b":"z e^z","option_c":"e^(z²)","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^z est e^z, comme en analyse réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"e^z\", \"b\": \"z e^z\", \"c\": \"e^(z²)\", \"d\": \"0\"}}","_debug_options_count":4},{"id":21608,"question":"Le module d'une fonction holomorphe est toujours constant.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le module d'une fonction holomorphe n'est pas nécessairement constant (exemple : f(z) = z).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]