Transforms and its Applications: UNIT 3: Fourier Series
Fourier Series: Half Range Sine Series
Fourier Series: Example Important Solved Problems with formula, steps, derivation, answer and Exercise Problems based on Half Range Sine Fourier Series.
Half‒Range Sine Series
Sine series: To expand f(x) as a sine series in (0, π) or (0,l), we extend the function reflecting it in the origin, so that f(x) = −f(x).
f(x) = g(x) in (0, π) or (0,1) is extended to f (x) = −g (−x) in (‒π, 0) or (‒l, 0).
Problems based on Half‒Range Sine Series
Example 1: Expand the function f(x) = x, 0 < x < π in Fourier sine series
Solution: Given f(x) = x in 0 < x < π
Example 2: Find the Half range Fourier sine series for f(x) = x in (0, l).
Solution:
Let the required Fourier series be
Example 3: Find the half range Fourier sine series for sinh ax in 0 < x < π.
Solution:
Given f(x) = sinh ax in 0 < x < π
Example 4: Find the half range sine series for the function f(x) = x − x2, 0 <x<1.
Solution:
Example 5: Find the sine series of f(x) = ex in (0, π).
Solution: Let the required Fourier series be
Example 6: Find the Fourier sine series of f(x) = 1−x in (0, l)
Solution:
Example 7: Find the Half range sine series for f(x) = x(π − x) in (0, π). Deduce that 1/13 ‒ 1/33 ‒ 1/53 ‒ …. = π3/32.
Solution: Let the half range sine series be
Example 8: Find the sine series of f(x) = 
Solution:
Let the half range sine series be
Example 9: Express f(x) as a Fourier sine series where
Solution:
Let the required Fourier sine series be
Example 10: Obtain the sine series for the function
Solution: The sine series for the function f(x) in (0, l) is given by
Example 11: If f(x)=k(lx − x2) in the range (0, l), show that the half range sine series for f(x) = 
Deduce that 1 ‒ 1/33 + 1/53 ‒ … = π3/32
Solution: Given: f(x) =k(lx − x2) in the range (0, l)
Transforms and its Applications: UNIT 3: Fourier Series : Tag: Engineering mathematics, Maths : - Fourier Series: Half Range Sine Series
Transforms and its Applications: UNIT 3: Fourier Series
Under Subject
Transforms and its Applications
MA25C03 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Related Subjects
English Essentials II
EN25C02 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Tamils and Technology தமிழர்களும் தொழில்நுட்பமும்
UC25H02 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Transforms and its Applications
MA25C03 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Applied Physics (EE) II
PH25C04 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Basic Civil and Mechanical Engineering
GE25C01 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation
Engineering Drawing
ME25C01 EEE, Mech, Agri, EEE Depts | 2025 Regulation | 2nd Semester 2025 Regulation
Data Structures and Algorithms
CS25C04 2nd Semester EEE Dept | 2025 Regulation
Re-Engineering for Innovation
ME25C05 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Engineering Drawing - Laboratory
ME25C01 2nd Semester | 2025 Regulation | 2nd Semester 2025 Regulation
Data Structures and Algorithms - Laboratory
CS25C04 2nd Semester EEE Dept | 2025 Regulation | 2nd Semester 2025 Regulation