Advanced Engineering Mathematics - Fourier Series

Explain trig functions with amplitude, period, phase shift, and vertical shift for sine and cosine; learn five-point plotting and analysis using y = a sin(bx+c)+d and y = (1/3) cos(2x+2/3).

Explore how amplitude, period, phase shift, and vertical shift affect sine and cosine graphs, compute starting points, plot points, and interpret one-cycle ranges for trigonometric functions.

Use the table of standard integrals to perform basic and advanced antiderivatives. Apply the power rule, integrate exponentials, square roots, sine and cosine, and one over x with the constant C.

Use integration by parts on integral from 0 to pi, x cos(n x) dx, with u = x and dv = cos(n x) dx, giving (cos(n pi) - 1)/n^2.

Apply integration by parts twice to evaluate a Fourier series coefficient, using limits 0 to pi and simplifying a product of two functions such as x and sin(nx) or cos(nx).

Dirichlet conditions require the function to be defined, single-valued, and periodic with period 2π. They require a finite number of discontinuities per period and Fourier series converge at continuity points.

Represent a piecewise periodic function as a Fourier series using a0, an, and bn; compute coefficients and build the infinite sine and cosine expansion, illustrated with a two-period example.

Derive Fourier series coefficients for a linear 2π-periodic function; get a0 = 6, a_n = 0, and b_n = ±6/π, yielding f(x) = 3 + (6/π) sum_{n=1}^ fty (-1)^n sin(nx).

identify whether a function is even, odd, or neither by examining its graph. note that evenness uses symmetry about the y-axis, while oddness uses symmetry about the origin.

Analyze a piecewise function on 0 to 2π that is not even nor odd, and derive its Fourier series by computing a0, an, and bn.

The lecture demonstrates constructing a half-range cosine series for a piecewise function on 0 to pi, extends it evenly, and derives the expansion f(x)=3/2+(2/π)[cos x−(1/2)cos2x+(1/3)cos3x−...], with b_n=0.

An introduction to half-range sine series for a piecewise function on [0, π], extending to an odd function and calculating b_n coefficients via integration by parts, with examples.

Compute a half-range sine series for f(x)=x^2 on [0,π], extend to [-π,π], and derive the sine-only Fourier series using symmetry and partial integration, as in example 2.

Demonstrates constructing a half-range cosine series for f(x)=sin x on [0, π] via even extension, calculating a0 and an (bn=0) and obtaining a cosine series with only even terms.

Derive the half-range sine series for f(x)=1+cos x on 0 to pi by odd extension and a trig identity, yielding the final 4/π ∑_{n=1}^∞ sin((2n−1)x)/(2n−1) with only odd harmonics.