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.

Describe periodic functions analytically from plots by constructing piecewise expressions, identifying the period, and applying linear or constant segment equations such as y = mx + b.

Develop skills to graph non-sinusoidal, periodic functions by plotting multiple cycles, labeling values, and using multiple points for piecewise, quadratic, and trigonometric examples.

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.

Explore evaluating definite integrals with symmetric bounds, using even/odd properties of sine and cosine, and trig identities to handle sin^2 x and cos^2 x, preparing for Fourier series coefficients.

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.

Explore advanced integration using integration by parts on x sin(nx) from 0 to pi, derive the resulting expression, and distinguish the even-integer case versus the general integer case for n.

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).

Apply the sin x cos y identity to convert the product to a sum, then split and evaluate the integral, yielding even n: 1/(1+n) + 1/(1−n); odd n: 0.

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.

Demonstrates calculating Fourier series coefficients for a two-interval piecewise function on 0 to 2 pi, using integration by parts to find a0, an, and bn, with plots.

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).

Compute Fourier series coefficients for a piecewise 2pi-periodic function, with x^2 on [0, pi] and 4 on [pi, 2pi], derive a0, a_n, b_n, and illustrate convergence.

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 whether functions are even or odd using analytic testing of f(-x), and apply to products, powers, and trig functions to classify examples.

Discover how the first Fourier theorem applies to even functions, yielding only cosine terms, with bn zero, and learn shortcuts using symmetry to compute a0 and an efficiently.

derive the cosine-only Fourier series for the even function x squared on [-pi, pi], yielding pi squared over three plus the series sum from n=1 to infinity of 4(-1)^n cos(nx)/n^2.

Shows that the even, piecewise function on the period-2π interval [-π, π] has a cosine-only Fourier series, computes a0 and odd-n coefficients, and presents the first terms of the series.

The second theorem for odd functions shows that an odd function on [−π, π] has a Fourier series with only sine terms; a0 and cosine coefficients vanish.

Identify the odd function on [-π, π], derive B_n = 24/(nπ) for odd n, and express f(x) = (24/π)[sin x + 1/3 sin 3x + 1/5 sin 5x + ...].

Demonstrate the second theorem for odd functions in Fourier series, showing only sine terms appear and presenting a shorter coefficient calculation via symmetry and integration.

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.

Introduce half-range cosine series for f(x)=2x on [0, pi], extend evenly to [-pi, pi], and derive coefficients a0 and an via integration in the Fourier series context.

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.