Euler’s homogeneous function theorem

Target entity: Euler’s homogeneous function theorem
Context triple: [Gibbs–Duhem equation, derivedFrom, Euler’s homogeneous function theorem]

• A. Euler’s theorem
  Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
• B. Shephard’s lemma
  Shephard’s lemma is a result in microeconomics stating that the derivative of a cost (or expenditure) function with respect to input (or price) yields the corresponding conditional factor (or Hicksian demand) demand function.
• C. Euler’s formula for complex exponentials
  Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
• D. Fuchs’ theorem
  Fuchs’ theorem is a result in the theory of linear differential equations that characterizes when all singular points of such an equation are regular (i.e., the equation is Fuchsian) based on the behavior of its coefficients.
• E. Euler’s reflection formula
  Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Euler’s homogeneous function theorem
Target entity description: Euler’s homogeneous function theorem is a fundamental result in multivariable calculus and thermodynamics that relates a homogeneous function to the sum of its variables times their partial derivatives, providing key constraints on extensive quantities.

• A. Euler’s theorem
  Euler’s theorem is a fundamental result in number theory stating that for any integer a coprime to n, a raised to the power of φ(n) is congruent to 1 modulo n.
• B. Shephard’s lemma
  Shephard’s lemma is a result in microeconomics stating that the derivative of a cost (or expenditure) function with respect to input (or price) yields the corresponding conditional factor (or Hicksian demand) demand function.
• C. Euler’s formula for complex exponentials
  Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
• D. Fuchs’ theorem
  Fuchs’ theorem is a result in the theory of linear differential equations that characterizes when all singular points of such an equation are regular (i.e., the equation is Fuchsian) based on the behavior of its coefficients.
• E. Euler’s reflection formula
  Euler’s reflection formula is a fundamental identity in complex analysis that relates the values of the Gamma function at z and 1−z through the sine function, revealing a deep symmetry of the Gamma function.
• F. None of above. chosen