introduction and Euler's formulae, conditions for a Fourier expansion and functions having points of discontinuity MCQs

1. A Fourier series represents a periodic function as:

a) A sum of exponential functions b) A sum of sine and cosine terms c) Either a or b d) A Taylor series expansion

Correct Answer: c) Either a or b

Explanation: A Fourier series represents a periodic function as a sum of sine and cosine terms in its trigonometric form or as a sum of exponential functions in its complex form using Euler's formula. Both representations are equivalent.

2. Euler's formula relates exponential and trigonometric functions as:

a) e^ix = cosx + isinx b) e^x = cosx + isinx c) e^ix = cosx - isinx d) e^x = cosx - isinx

Correct Answer: a) e^ix = cosx + isinx

Explanation: Euler's formula is a fundamental relationship in mathematics, given by: $e^{ix} = \cos (x) + i \sin (x)$ where:

$e$ is the base of the natural logarithm.
$i$ is the imaginary unit ( $i = \sqrt{- 1}$ ).
$x$ is a real number.

This formula connects complex exponentials with trigonometric functions and is widely used in fields like signal processing, quantum mechanics, and electrical engineering.

3. The Dirichlet conditions require that a periodic function must:

a) Be continuous everywhere b) Have a finite number of extrema c) Be differentiable everywhere d) Have infinite discontinuities

Correct Answer: b) Have a finite number of extrema

Explanation: Dirichlet conditions require: finite discontinuities, finite extrema, and absolute integrability over one period.

4. The coefficient a₀ in a Fourier series represents:

a) The fundamental frequency b) The average value of the function c) The highest harmonic d) The phase shift

Correct Answer: b) The average value of the function

Explanation: In a Fourier series, the coefficient $a_{0}$ represents the average value of the function over one period. It is calculated as: $a_{0} = \frac{1}{T} \int_{0}^{T} f (t) d t$ where $T$ is the period of the function. This coefficient represents the constant or DC component of the function in its Fourier series expansion.

5. At a point of discontinuity, the Fourier series converges to:

a) The left-hand limit b) The right-hand limit c) The average of left and right limits d) Zero

Correct Answer: c) The average of left and right limits

Explanation: At discontinuities, the series converges to the average of the left and right limits (f(x⁺) + f(x^-))/2.

6. The Fourier series of an odd function contains only:

a) Cosine terms b) Sine terms c) Both sine and cosine terms d) Constant term

Correct Answer: b) Sine terms

Explanation: Odd functions have only sine terms in their Fourier expansion (all a_n coefficients are zero).

7. The Gibbs phenomenon refers to:

a) Slow convergence at continuous points b) Overshoot near discontinuities c) Missing harmonics d) Phase shifts in coefficients

Correct Answer: b) Overshoot near discontinuities

Explanation: Gibbs phenomenon is the peculiar overshoot that occurs near discontinuities in Fourier series approximations.

8. The Fourier series representation of a function is always:

a) Continuous b) Differentiable c) Periodic d) Analytic

Correct Answer: c) Periodic

Explanation: Fourier series representations are inherently periodic, repeating over the fundamental period.

9. A piecewise continuous function has:

a) No discontinuities b) Infinite discontinuities c) Finite jump discontinuities d) Continuous derivatives

Correct Answer: c) Finite jump discontinuities

Explanation: Piecewise continuous functions have a finite number of jump discontinuities and are continuous between them.

10. At points where a function is continuous, its Fourier series:

a) May not converge b) Converges to the function value c) Always diverges d) Converges to zero

Correct Answer: b) Converges to the function value

Explanation: At points of continuity, the Fourier series converges pointwise to the original function value.

Introduction and Euler's formulae, conditions for a Fourier expansion and functions having points of discontinuity

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