Studdy AI Logo
Math  /  Algebra

QuestionExpress the following polynomial function in terms of Legendre Polynomial f(x)=x3+1f(x)=x^{3}+1

Studdy Solution

STEP 1

1. We are given the polynomial function f(x)=x3+1 f(x) = x^3 + 1 .
2. We need to express this function in terms of Legendre polynomials.

STEP 2

1. Understand the properties of Legendre polynomials.
2. Express the given polynomial as a linear combination of Legendre polynomials.
3. Calculate the coefficients for each Legendre polynomial.
4. Write the final expression of the polynomial in terms of Legendre polynomials.

STEP 3

Understand the properties of Legendre polynomials:
Legendre polynomials Pn(x) P_n(x) are orthogonal polynomials defined on the interval [−1,1][-1, 1] with the orthogonality condition:
∫−11Pm(x)Pn(x) dx=0form≠n\int_{-1}^{1} P_m(x) P_n(x) \, dx = 0 \quad \text{for} \quad m \neq n
The first few Legendre polynomials are: - P0(x)=1 P_0(x) = 1 - P1(x)=x P_1(x) = x - P2(x)=12(3x2−1) P_2(x) = \frac{1}{2}(3x^2 - 1) - P3(x)=12(5x3−3x) P_3(x) = \frac{1}{2}(5x^3 - 3x)

STEP 4

Express the given polynomial as a linear combination of Legendre polynomials:
We want to express f(x)=x3+1 f(x) = x^3 + 1 as:
f(x)=c0P0(x)+c1P1(x)+c2P2(x)+c3P3(x)f(x) = c_0 P_0(x) + c_1 P_1(x) + c_2 P_2(x) + c_3 P_3(x)

STEP 5

Calculate the coefficients cn c_n using the orthogonality condition:
The coefficients are given by:
cn=2n+12∫−11f(x)Pn(x) dxc_n = \frac{2n + 1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx
Calculate each coefficient:
1. c0 c_0 :
c0=12∫−11(x3+1)⋅1 dx=12[∫−11x3 dx+∫−111 dx]c_0 = \frac{1}{2} \int_{-1}^{1} (x^3 + 1) \cdot 1 \, dx = \frac{1}{2} \left[ \int_{-1}^{1} x^3 \, dx + \int_{-1}^{1} 1 \, dx \right]
2. c1 c_1 :
c1=32∫−11(x3+1)⋅x dx=32[∫−11x4 dx+∫−11x dx]c_1 = \frac{3}{2} \int_{-1}^{1} (x^3 + 1) \cdot x \, dx = \frac{3}{2} \left[ \int_{-1}^{1} x^4 \, dx + \int_{-1}^{1} x \, dx \right]
3. c2 c_2 :
c2=52∫−11(x3+1)⋅12(3x2−1) dxc_2 = \frac{5}{2} \int_{-1}^{1} (x^3 + 1) \cdot \frac{1}{2}(3x^2 - 1) \, dx
4. c3 c_3 :
c3=72∫−11(x3+1)⋅12(5x3−3x) dxc_3 = \frac{7}{2} \int_{-1}^{1} (x^3 + 1) \cdot \frac{1}{2}(5x^3 - 3x) \, dx

STEP 6

Evaluate each integral to find the coefficients:
1. c0 c_0 :
c0=12[x44∣−11+x∣−11]=12[0+2]=1c_0 = \frac{1}{2} \left[ \left. \frac{x^4}{4} \right|_{-1}^{1} + \left. x \right|_{-1}^{1} \right] = \frac{1}{2} \left[ 0 + 2 \right] = 1
2. c1 c_1 :
c1=32[x55∣−11+x22∣−11]=32[0+0]=0c_1 = \frac{3}{2} \left[ \left. \frac{x^5}{5} \right|_{-1}^{1} + \left. \frac{x^2}{2} \right|_{-1}^{1} \right] = \frac{3}{2} \left[ 0 + 0 \right] = 0
3. c2 c_2 :
c2=52[12(∫−113x5−x3 dx+∫−113x2−1 dx)]=0c_2 = \frac{5}{2} \left[ \frac{1}{2} \left( \int_{-1}^{1} 3x^5 - x^3 \, dx + \int_{-1}^{1} 3x^2 - 1 \, dx \right) \right] = 0
4. c3 c_3 :
c3=72[12(∫−115x6−3x4 dx+∫−115x3−3x dx)]=1c_3 = \frac{7}{2} \left[ \frac{1}{2} \left( \int_{-1}^{1} 5x^6 - 3x^4 \, dx + \int_{-1}^{1} 5x^3 - 3x \, dx \right) \right] = 1

STEP 7

Write the final expression of the polynomial in terms of Legendre polynomials:
f(x)=1⋅P0(x)+0⋅P1(x)+0⋅P2(x)+1⋅P3(x)f(x) = 1 \cdot P_0(x) + 0 \cdot P_1(x) + 0 \cdot P_2(x) + 1 \cdot P_3(x)
f(x)=P0(x)+P3(x)f(x) = P_0(x) + P_3(x)
f(x)=1+12(5x3−3x)f(x) = 1 + \frac{1}{2}(5x^3 - 3x)
The expression of the polynomial in terms of Legendre polynomials is:
f(x)=1+12(5x3−3x)f(x) = 1 + \frac{1}{2}(5x^3 - 3x)

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord