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Choose the function f(t); –∞ < t < ∞, for which a Fourier series cannot be defined.
- a)3 sin (25 t)
- b)4 cos (20 t + 3) + 2 sin (710 t)
- c)exp (–|t|) sin (25 t)
- d)1
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Choose the function f(t); t , for which a Fourier series cannot be ...
Explanation:
To determine whether a Fourier series can be defined for a function f(t), we need to check if the function satisfies the Dirichlet conditions. The Dirichlet conditions are:
1. The function must be periodic with a period T.
2. The function must have a finite number of maxima and minima in any finite interval.
3. The function must have a finite number of discontinuities in any finite interval.
4. The integral of the absolute value of the function over a period must be finite.
If a function satisfies all of these conditions, then its Fourier series can be defined. If any of these conditions are not satisfied, then the Fourier series cannot be defined.
Option C: exp(|t|)sin(25t)
This function is not periodic since its amplitude increases exponentially as t approaches infinity. Therefore, it does not satisfy the first Dirichlet condition and a Fourier series cannot be defined for this function.
To determine whether a Fourier series can be defined for a function f(t), we need to check if the function satisfies the Dirichlet conditions. The Dirichlet conditions are:
1. The function must be periodic with a period T.
2. The function must have a finite number of maxima and minima in any finite interval.
3. The function must have a finite number of discontinuities in any finite interval.
4. The integral of the absolute value of the function over a period must be finite.
If a function satisfies all of these conditions, then its Fourier series can be defined. If any of these conditions are not satisfied, then the Fourier series cannot be defined.
Option C: exp(|t|)sin(25t)
This function is not periodic since its amplitude increases exponentially as t approaches infinity. Therefore, it does not satisfy the first Dirichlet condition and a Fourier series cannot be defined for this function.
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Community Answer
Choose the function f(t); t , for which a Fourier series cannot be ...
Fourier series is defined for periodic function and constant.
Fourier series can’t be defined for non-periodic functions.
The sum of two periodic function is also a periodic function.
3 sin (25 t) is a periodic function
4 cos (20 t + 3) and 2 sin (710 t) both are periodic function. So, 4 cos (20t + 3) + 2 sin (710 t) is also a periodic function.
e-|t| sin (25 t) is not a periodic function. So, Fourier series is can’t be defined.
1 is a constant. So, Fourier series is defined.
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