Give the number to which the fourier series converges at a point of discontinuity of f.

f(x) = 3 - 2x, -π < x < π

The Fourier series of f(x) converges to 3 at the points x= π+2kπ, where k is an integer.

Step-by-step explanation:

First, recall that the function f(x) is extended 2π periodic to the whole real line, in order to obtain a valid Fourier expansion. Remember that a Fourier series is formed by a sines and cosines, which are 2π-periodic.

So, the 2π-periodic expansion of f(x) is discontinuous at the points π+2kπ, in particular π and -π. Check the attached figure to a better understanding.

Now, the Dirichlet theorem on the convergence of a Fourier series tells us that the series converges to the function at the points of continuity, and at points of discontinuity the sum of the series is

.

Here we understand the notation and as

and

.

In this particular case

.

For the limit , with recall that our function is 2π-periodic, so the values of f near π, with x>π are the same when x is near -π and x>-π. Again, check the attached figure. So,

.

Thus,

.

Note: In the attached figure we only have drawn three repetitions of the 2π-periodic extension of , recall that the extension is ad infinitum. Also, the points drawn in the dotted lines are the sum of the series at the points of discontinuity.

this story was prepared by jim

mccullaugh in los angeles and

al stewart in new york.

new york the colossal success of "top gun" appears to have dramatically altered madison avenue's

perception of corporate sponsorship

of home video projects.

the paramount home video re- lease, featuring a 60- second diet

pepsi spot, was by no means the

first prerecorded video to carry an advertiser's plug, but it was the

first major theatrical release to do so. in the aftermath of the title's

success, paramount is widely ex- pected to ink a similar deal for

"crocodile dundee" or "star trek

iv" or both.

corporate america's new attraccds push british

recording sales

to 19% gain

london fueled by the compact

disk boom, the value of u.k. manu- facturer shipments of recordings

rose 19% in the 12 -month period ended march 1987, according to the

british phonographic industry.

total value of u.k. shipments for

the period was $755 million for all

configurations. (all value figures

are based on an exchange rate of

$1.68 to the pound sterling.)

cd sales totaled 10.7 million units

worth $121.3 million- compared

with 3.7 million units worth $39.5

million in the prior 12 months -and

accounted for two -thirds of the

overall gain in trade value. at present growth rates, says the bpi, cd

sales for calendar 1987 will reach 20

(continued on page 79)

step-by-step explanation:

1 + (6 / x - 1)

step-by-step explanation:

divide using polynomial division.