Evaluation

Input	Output
What does the following represent: $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$	Anthropic: Claude 3 Haiku PublicPass C The expression $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$ represents the definite integral of the Gaussian or normal distribution function over the entire real line, which is a fundamental result in mathematics and probability theory. Here's what the expression means: $\int_{-\infty}^\infty$ indicates that the integral is taken over the entire real line, from negative infinity to positive infinity. $e^{-x^2}$ is the Gaussian or normal distribution function, which is a fundamental probability density function in probability and statistics. This function has the shape of a bell-shaped curve, and it decays rapidly as $\|x\|$ increases. $\,dx$ is the differential element, indicating that the integral is taken with respect to the variable $x$ . The result of this integral is $\sqrt{\pi}$ , which is a constant with an approximate value of 1.7724538509055160272981674833411. This result is important in many areas of mathematics and physics, as the Gaussian function and its integral appear in various contexts, such as: Probability theory: The Gaussian distribution is a widely used probability distribution for modeling many natural and experimental phenomena. Quantum mechanics: The Gaussian function appears in the solutions to the Schrödinger equation for various quantum systems. Fourier analysis: The Gaussian function is the fundamental solution to the heat equation and is closely related to the Fourier transform. Signal processing: The Gaussian function is used to design filters and windows for signal processing applications. The fact that the integral of the Gaussian function over the entire real line is equal to $\sqrt{\pi}$ is a fundamental result that has many important applications in various fields of science and mathematics.

Input

Output

What does the following represent: $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$

PublicPass

The expression $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$ represents the definite integral of the Gaussian or normal distribution function over the entire real line, which is a fundamental result in mathematics and probability theory.

Here's what the expression means:

$\int_{-\infty}^\infty$ indicates that the integral is taken over the entire real line, from negative infinity to positive infinity.
$e^{-x^2}$ is the Gaussian or normal distribution function, which is a fundamental probability density function in probability and statistics. This function has the shape of a bell-shaped curve, and it decays rapidly as $|x|$ increases.
$\,dx$ is the differential element, indicating that the integral is taken with respect to the variable $x$ .
The result of this integral is $\sqrt{\pi}$ , which is a constant with an approximate value of 1.7724538509055160272981674833411.

This result is important in many areas of mathematics and physics, as the Gaussian function and its integral appear in various contexts, such as:

Probability theory: The Gaussian distribution is a widely used probability distribution for modeling many natural and experimental phenomena.
Quantum mechanics: The Gaussian function appears in the solutions to the Schrödinger equation for various quantum systems.
Fourier analysis: The Gaussian function is the fundamental solution to the heat equation and is closely related to the Fourier transform.
Signal processing: The Gaussian function is used to design filters and windows for signal processing applications.

The fact that the integral of the Gaussian function over the entire real line is equal to $\sqrt{\pi}$ is a fundamental result that has many important applications in various fields of science and mathematics.

Evaluation ID: 2e7cb060-bdec-436e-80c9-4a7340697e93
Created: April 23, 2024

Evaluation Rating

Binary Pass/Fail: Pass

Score: 100

Factual Consistency: C: the submitted answer contains all the same details as the expert answer

Raw Output

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