How is the Sine Series Derived for Piecewise Functions?

In summary, the conversation discussed the sine series for a given function and how it was obtained using integration. The final result was a series with different N values that represented the function in terms of sine and cosine. The conversation also mentioned the use of different values for a and b to calculate the series.
  • #1
hawaiidude
41
0
why is

the sine series for f(x) = { 0 x<2 on 0,3
2 x>2

L=3

cn=2/3 int b=3 and a =0 f(x) sin( nxpi/ L)dx
= 2/3 int. b=2 a=0 (0) sin( pi n x/ 3) dx + 2/3 int b=3 a =2 (2) sin n x pi/3 dx
= 0 + 4/3{-3n pi cos n pi x/3} x=3 and x=2 = 4/n pi { cos 2 n pi/3 - cos n pi }

f(x)=sigma n=1 4/n pi { cos 2n pi/3 -(-1)^n sin n pi x

cos 2 pi/3=-1/2...

f(x)=4/pi( 2/3 sin pix/3-3/4sin2 pi x/3 +2/3sin 3 pi x/3...

how did they get this? very confused
 
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  • #2
my guess would be differnt N values
 
  • #3
ok never mind i get it now
 

FAQ: How is the Sine Series Derived for Piecewise Functions?

What is a sine series?

A sine series is a mathematical representation of a periodic function using an infinite series of sine functions. It is often used to approximate a function or to find the values of a function at various points.

How is the sine series for f(x) = { 0 x<2 } calculated?

The sine series for f(x) = { 0 x<2 } can be calculated using the formula: f(x) = a0/2 + Σ(an*cos(nx) + bn*sin(nx)), where a0 is the average value of the function, an and bn are coefficients, and n is the number of terms in the series.

What is the purpose of using a sine series for f(x) = { 0 x<2 }?

The purpose of using a sine series for f(x) = { 0 x<2 } is to approximate the function and find its values at different points. It can also help in understanding the behavior of the function and identifying any patterns.

Can the sine series for f(x) = { 0 x<2 } be used for any other functions?

Yes, the sine series for f(x) = { 0 x<2 } can be used for any periodic function, i.e. a function that repeats itself after a certain interval. It can also be used to approximate non-periodic functions with a certain degree of accuracy.

How many terms are needed to get an accurate representation of f(x) = { 0 x<2 } using a sine series?

The number of terms needed for an accurate representation of f(x) = { 0 x<2 } using a sine series depends on the desired level of accuracy. Generally, the more terms used, the more accurate the representation will be. However, it is common to use only a few terms for simpler functions like f(x) = { 0 x<2 }.

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