It is impossible within a finite amount of time to add an infinitely- numerous series of terms sequentially (one after the other) unless you can already deduce the answer, so this Fourier Series might seem like it isn't very useful, however it is quite the opposite. Physicists often prove that such a series converges to a finite limit at all function values of interest, that consecutive terms in the sum are less and less significant, and then they approximate a physical situation by using the first few terms, or perhaps 100 terms or 1000 terms etcetera, depending on the precision they need. Use a computer program such as Microsoft Excel to plot on the same set of axes, and on the interval given, the function f(x) = x PHY-2112 Duplication or distribution prohibited without express written consent from Stephen Fahey. Instructor: Stephen Fahey well as several approximations to the Fourier Series expansion with 2 " as only the first 1 term n=1 the first 2 terms n=1, the first 3 terms, and the first 10 terms. Thus, your final answer for this question should have 5 trends plotted on one set of axes. If you plot individual points then choose a point-spacing small-enough to see all key features of each trend.
It is impossible within a finite amount of time to add an infinitely- numerous series of terms sequentially (one after the other) unless you can already deduce the answer, so this Fourier Series might seem like it isn't very useful, however it is quite the opposite. Physicists often prove that such a series converges to a finite limit at all function values of interest, that consecutive terms in the sum are less and less significant, and then they approximate a physical situation by using the first few terms, or perhaps 100 terms or 1000 terms etcetera, depending on the precision they need. Use a computer program such as Microsoft Excel to plot on the same set of axes, and on the interval given, the function f(x) = x PHY-2112 Duplication or distribution prohibited without express written consent from Stephen Fahey. Instructor: Stephen Fahey well as several approximations to the Fourier Series expansion with 2 " as only the first 1 term n=1 the first 2 terms n=1, the first 3 terms, and the first 10 terms. Thus, your final answer for this question should have 5 trends plotted on one set of axes. If you plot individual points then choose a point-spacing small-enough to see all key features of each trend.
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Transcribed Image Text:It is impossible within a finite amount of time to add an infinitely-
numerous series of terms sequentially (one after the other) unless
you can already deduce the answer, so this Fourier Series might seem
like it isn't very useful, however it is quite the opposite. Physicists
often prove that such a series converges to a finite limit at all
function values of interest, that consecutive terms in the sum are
less and less significant, and then they approximate a physical
situation by using the first few terms, or perhaps 100 terms or 1000
terms etcetera, depending on the precision they need.
Use a computer program such as Microsoft Excel to plot on the
same set of axes, and on the interval given, the function f(x)=x
PHY-2112 Duplication or distribution prohibited without express written consent from Stephen Fahey. Instructor: Stephen Fahey
well as several approximations to the Fourier Series expansion with
2
as
only the first 1 term n=1 the first 2 terms n=1 the first 3 terms,
and the first 10 terms. Thus, your final answer for this question
should have 5 trends plotted on one set of axes. If you plot
individual points then choose a point-spacing small-enough to see all
key features of each trend.
"
![4) [swHW] It turns out that any function that has a finite number of
finite-magnitude discontinuities and a finite number of extrema
(maximums and minimums) over a finite interval can be represented
exactly with an infinite series of cosine and sine functions called a
Fourier Series. The conditions that the function must meet, called
the "Dirichlet conditions", are not very restrictive, so most functions
you will encounter in physics will have an associated Fourier Series.
To give you a sense of how this is possible consider a very simple
f(x) = x
function
on the interval -π<x<π, which seems very
different from cosine or sine in form. The Fourier Series for this
function on the given interval is
1
f(x)=2 Σ(-1)"+
Σ(-1)"+¹=sin(nx)
n
n=1
∞](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd295e1c3-483d-49b7-8dbf-fcde864060f6%2Fb4fc0765-fb70-4ec4-8f31-13fba57321f6%2Fmbm18z_processed.png&w=3840&q=75)
Transcribed Image Text:4) [swHW] It turns out that any function that has a finite number of
finite-magnitude discontinuities and a finite number of extrema
(maximums and minimums) over a finite interval can be represented
exactly with an infinite series of cosine and sine functions called a
Fourier Series. The conditions that the function must meet, called
the "Dirichlet conditions", are not very restrictive, so most functions
you will encounter in physics will have an associated Fourier Series.
To give you a sense of how this is possible consider a very simple
f(x) = x
function
on the interval -π<x<π, which seems very
different from cosine or sine in form. The Fourier Series for this
function on the given interval is
1
f(x)=2 Σ(-1)"+
Σ(-1)"+¹=sin(nx)
n
n=1
∞
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