Question
फलन का प्रसार
के रूप की श्रेणी में करें।
Answer :
Word Count : 536
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फलन (f(x) = x^4 - 1) को लेजेंड्रे बहुपद (P_k(x)) के रूप में व्यक्त करने के लिए, हमें इसे लेजेंड्रे बहुपद के आधार पर फैलाना होगा। लेजेंड्रे बहुपद (P_0(x), P_1(x), P_2(x), P_3(x), P_4(x), \dots) निम्नलिखित हैं: [ \begin{aligned} P_0(x) &= 1 \ P_1(x) &= x \ P_2(x) &= \frac{1}{2}(3x^2 - 1) \ P_3(x) &= \frac{1}{2}(5x^3 - 3x) \ P_4(x) &= \frac{1}{8}(35x^4 - 30x^2 + 3) \end{aligned} ] हम जानते हैं कि लेजेंड्रे बहुपद पर प्रसार: [ f(x) = \sum_{k=0}^{\infty} A_k P_k(x), \quad \text{जहाँ } A_k = \frac{2k+1}{2} \int_{-1}^{1} f(x) P_k(x) , dx ] चूंकि (f(x) = x^4 - 1) एक सम फ़ंक्शन है, सभी विषम (k) के लिए (A_k = 0) होंगे। अतः केवल (k = 0,2,4) पर ध्यान देंगे। चरण 1: (A_0) की गणना [ A_0 = \frac{1}{2} \int_{-1}^{1} (x^4 - 1) \cdot 1 , dx = \frac{1}{2} \int_{-1}^{1} (x^4 - 1) , dx ] [ \int_{-1}^{1} x^4 dx = \left[ \frac{x^5}{5} \right]_{-1}^{1} = \frac{1}{5} - \left(-\frac{1}{5}\right) = \frac{2}{5} ] [ \int_{-1}^{1} 1 , dx = 2 ] इसलिए: [ \int_{-1}^{1} (x^4 - 1) dx = \frac{2}{5} - 2 = -\frac{8}{5} ] तो: [ A_0 = \frac{1}{2}
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