Fourier Series Properties for Even and Odd Functions
The composition of a Fourier series expansion depends critically on the symmetry properties of the function being represented.
Even Functions and Fourier Series
A function '$f(x)$' is considered even if it satisfies the property '$f(x) = f(-x)$' for all '$x$'. For such functions, their Fourier series expansion consists solely of a constant term and cosine terms.
Mathematically, the coefficients for the sine terms '$b_n$' are zero for an even function.
Odd Functions and Fourier Series
A function '$f(x)$' is defined as odd if it adheres to the condition '$f(x) = -f(-x)$' for all '$x$'. The Fourier series expansion of an odd function includes only sine terms.
Mathematically, the coefficients for the cosine terms '$a_0$' and '$a_n$' are zero for an odd function.
Evaluating the Provided Statements
- Statement 1: "The Fourier series of an even function contains only cosine terms" - This aligns with the property that even functions result in a Fourier series with only cosine components (and the constant term). Correct.
- Statement 2: "The Fourier series of an odd function contains only cosine terms" - This is incorrect, as odd functions yield sine terms.
- Statement 3: "The Fourier series of an odd function contains only sine terms" - This correctly describes the Fourier series expansion for odd functions. Correct.
- Statement 4: "The Fourier series of an even function contains only sine terms" - This is incorrect, as even functions yield cosine terms.
Based on this analysis, the correct statements are the first and the third.
Important Questions from Fourier Series
If we use the Fourier transform ϕ(x, y) = \(\int {{{\rm{e}}^{{\rm{ikx}}}}} {ϕ _{\rm{k}}}\left( {\rm{y}} \right){\rm{dk}}\) to solve the partial differential equation \({\rm{ - }}\frac{{{\partial ^2}ϕ \left( {x,y} \right)}}{{\partial {y^2}}}\, - \,\frac{1}{{{y^2}}}\frac{{{\partial ^2}ϕ \left( {x,y} \right)}}{{\partial {x^2}}} + \frac{{{m^2}}}{{{y^2}}}ϕ \left( {x,y} \right) = 0\) in the half-plane {(x, y) : -∞ < x < ∞, 0 < y < ∞} the Fourier modes ϕ k(y) depend on y as y α and y β . The values of α and β are
When a time-domain signal is converted into its Fourier representation, which of the following is/are conserved?
I. Energy
II. Power
The trigonometric Fourier series of a periodic time function can have
The Fourier series expansion of x3 in the interval −1 ≤ x < 1 with periodic continuation has
The Fourier series to represent x-x2 for –π ≤ x ≤ π is given by \(x - {x^2} = \frac{{{a_0}}}{2} + \mathop \sum \limits_{n = 1}^\infty {a_n}cosnx + \mathop \sum \limits_{n = 1}^\infty {b_n}sinnx\)
The value of a0 (round off to two decimal places), is
