Quiz — 5-Fonction Digamma .pdf

Question 1 sur 10 20:00

[{"id":64896,"question":"Quelle est la relation entre la fonction digamma ψ(x) et la fonction gamma Γ(x) ?","option_a":"ψ(x) = Γ'(x)\/Γ(x)","option_b":"ψ(x) = Γ(x)\/Γ'(x)","option_c":"ψ(x) = d\/dx [ln(Γ(x))]","option_d":"ψ(x) = Γ(x) * Γ'(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction digamma est définie comme la dérivée logarithmique de la fonction gamma : ψ(x) = d\/dx [ln(Γ(x))] = Γ'(x)\/Γ(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ψ(x) = Γ'(x)\/Γ(x)\", \"b\": \"ψ(x) = Γ(x)\/Γ'(x)\", \"c\": \"ψ(x) =","_debug_options_count":4},{"id":64897,"question":"La fonction digamma ψ(x) est-elle définie pour x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction digamma ψ(x) n'est pas définie pour x = 0 car Γ(0) n'est pas défini (pôle de la fonction gamma).","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":64898,"question":"Quelle est la valeur de ψ(1) ?","option_a":"-1","option_b":"0","option_c":"-γ (constante d'Euler-Mascheroni)","option_d":"1","option_e":"","option_f":"","bonne_reponse":"c","explication":"ψ(1) = -γ, où γ ≈ 0.5772 est la constante d'Euler-Mascheroni.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"-1\", \"b\": \"0\", \"c\": \"-γ (constante d'Euler-Mascheroni)\", \"d\": \"1","_debug_options_count":4},{"id":64899,"question":"La fonction digamma satisfait-elle la relation de récurrence ψ(x+1) = ψ(x) + 1\/x ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, la fonction digamma vérifie bien la relation de récurrence ψ(x+1) = ψ(x) + 1\/x pour tout x ≠ 0, -1, -2, ...","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":64900,"question":"Quelle est la limite de ψ(x) lorsque x tend vers l'infini ?","option_a":"0","option_b":"1","option_c":"ln(x) - 1\/(2x) + ...","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour x grand, ψ(x) ≈ ln(x) - 1\/(2x) - 1\/(12x²) + ..., ce qui montre que ψ(x) tend vers l'infini comme ln(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"ln(x) - 1\/(2x) + ...\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":64901,"question":"La fonction digamma est-elle une fonction périodique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, la fonction digamma n'est pas périodique. Elle tend vers l'infini lorsque x tend vers l'infini et présente des pôles en x = 0, -1, -2, ...","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":64902,"question":"Quelle est la dérivée de la fonction digamma ψ(x) ?","option_a":"ψ'(x) = ψ(x)","option_b":"ψ'(x) = Γ''(x)\/Γ(x)","option_c":"ψ'(x) = ψ²(x)","option_d":"ψ'(x) = Γ'(x)\/Γ(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de la fonction digamma est appelée fonction trigamma : ψ'(x) = d²\/dx² [ln(Γ(x))] = Γ''(x)\/Γ(x) - (Γ'(x)\/Γ(x))².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"ψ'(x) = ψ(x)\", \"b\": \"ψ'(x) = Γ''(x)\/Γ(x)\", \"c\": \"ψ'(x) = ψ","_debug_options_count":4},{"id":64903,"question":"La fonction digamma est-elle convexe sur son domaine de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, la fonction digamma est convexe sur son domaine de définition car sa dérivée seconde (fonction trigamma) est positive pour x \u003E 0.","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":64904,"question":"Quelle est la valeur de ψ(2) en fonction de ψ(1) ?","option_a":"ψ(2) = ψ(1) + 1","option_b":"ψ(2) = ψ(1) - 1","option_c":"ψ(2) = 2ψ(1)","option_d":"ψ(2) = ψ(1)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la relation de récurrence ψ(x+1) = ψ(x) + 1\/x, on obtient ψ(2) = ψ(1) + 1\/1 = ψ(1) + 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"ψ(2) = ψ(1) + 1\", \"b\": \"ψ(2) = ψ(1) - 1\", \"c\": \"ψ(2) = 2ψ(1","_debug_options_count":4},{"id":64905,"question":"La fonction digamma peut-elle s'exprimer comme une série infinie ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, pour x \u003E 0, ψ(x) = -γ + ∑(k=0 à ∞) [1\/(k+1) - 1\/(k+x)], où γ est la constante d'Euler-Mascheroni.","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}]