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) (A)</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 (A)</p> Signup and view all the answers

Who studied the beta function and gave it its name?

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

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

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

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

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

What is a key property of the beta function?

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

What is the integral representation of the beta function?

<p>$B(z_1, z_2) = \int_0^1 t^{z_1-1} (1-t)^{z_2-1} dt$ (D)</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 $\Re(z_{1}), \Re(z_{2}) &gt; 0$ (B)</p> Signup and view all the answers

Who gave the beta function its name?

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

What is a key property of the beta function?

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

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

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

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

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

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