The Beta Function Quiz
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Questions and Answers

What is the definition of the beta function?

  • B(z1, z2) = ∫ 0 1 t^{z1}(1-t)^{z2} dt
  • B(z1, z2) = ∫ 0 1 t^{z1-1}(1-t)^{z2} dt
  • B(z1, z2) = ∫ 0 1 t^{z1}(1-t)^{z2-1} dt
  • B(z1, z2) = ∫ 0 1 t^{z1-1}(1-t)^{z2-1} dt (correct)

    • Who gave the beta function its name?

  • Pierre-Simon Laplace
  • Adrien-Marie Legendre
  • Jacques Binet (correct)
  • Leonhard Euler

    • What is a key property of the beta function?

  • Symmetry: B(z1, z2) = B(z2, z1) (correct)
  • Transitivity: B(z1, z2) = B(z1, z3)
  • Asymmetry: B(z1, z2) ≠ B(z2, z1)
  • Reflexivity: B(z1, z1) = 1

    • What is the relationship between the beta function and the gamma function?

    <p>B(z1, z2) = Γ(z1)Γ(z2)</p> Signup and view all the answers

    Under what conditions is the beta function defined for complex number inputs z1, z2?

    <p>ℜ(z1) &gt; 0, ℜ(z2) &gt; 0</p> Signup and view all the answers

    Who studied the beta function and gave it its name?

    <p>Leonhard Euler and Adrien-Marie Legendre</p> Signup and view all the answers

    What is the relationship between the beta function and the gamma function?

    <p>$B(z_1, z_2) = \Gamma(z_1) \Gamma(z_2)$</p> Signup and view all the answers

    Under what conditions is the beta function defined for complex number inputs $z_1, z_2$?

    <p>$\Re(z_1), \Re(z_2) &gt; 0$</p> Signup and view all the answers

    What is a key property of the beta function?

    <p>Symmetry: $B(z_1, z_2) = B(z_2, z_1)$</p> Signup and view all the answers

    What is the integral representation of the beta function?

    <p>$B(z_1, z_2) = \int_0^1 t^{z_1-1} (1-t)^{z_2-1} dt$</p> Signup and view all the answers

    What is the definition of the beta function?

    <p>$B(z_{1}, z_{2}) = \int_{0}^{1} t^{z_{1}-1}(1-t)^{z_{2}-1} dt$ for $\Re(z_{1}), \Re(z_{2}) &gt; 0$</p> Signup and view all the answers

    Who gave the beta function its name?

    <p>Jacques Binet</p> Signup and view all the answers

    What is a key property of the beta function?

    <p>Symmetry: $B(z_{1}, z_{2}) = B(z_{2}, z_{1})$</p> Signup and view all the answers

    What is the relationship between the beta function and the gamma function?

    <p>$B(z_{1}, z_{2}) = \Gamma(z_{1})\Gamma(z_{2})$</p> Signup and view all the answers

    Under what condition is the beta function defined for complex number inputs $z_{1}, z_{2}$?

    <p>$\Re(z_{1}), \Re(z_{2}) &gt; 0$</p> Signup and view all the answers

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