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-   -   odd or even function (http://mymathforum.com/complex-analysis/341585-odd-even-function.html)

aows61 August 27th, 2017 02:17 PM

odd or even function
 
Hello dear,

I have a question in complex Fourier series:
Q:/ how do I know whether the function is even or odd? if after solving:
a$_0$ = 2$\pi$
a$_n$ = 0
b$_n$ = -2/$\pi$
C$_n$ = j/n
C$_{-n}$ = -j/n
Can you help me with that?

skipjack August 27th, 2017 06:05 PM

Whether it's complex is irrelevant. Write the nth term (for non-zero n) in terms of sin and cos. For an odd function, the non-zero terms are all sin terms. For an even function, there are no non-zero sin terms.

aows61 August 28th, 2017 02:35 AM

clarify
 
Quote:

Originally Posted by skipjack (Post 578901)
Whether it's complex is irrelevant. Write the nth term (for non-zero n) in terms of sin and cos. For an odd function, the non-zero terms are all sin terms. For an even function, there are no non-zero sin terms.

Hello Dear,
can you clarify more ?

Joppy August 28th, 2017 03:50 AM

Quote:

Originally Posted by aows61 (Post 578931)
Hello Dear,

Something is not quite right here :lol:

aows61 August 28th, 2017 03:54 AM

share answer
 
Quote:

Originally Posted by Joppy (Post 578934)
Something is not quite right here :lol:

Hello Dear,
can you share your answer?

Joppy August 28th, 2017 04:02 AM

Quote:

Originally Posted by aows61 (Post 578935)
Hello Dear,
can you share your answer?

Sorry, i was actually referring to your use of the word 'dear'. Typically we use it when writing a formal letter. For example, "Dear aows61, how are you today?".

However use of 'dear' in the context here ('hello dear') refers to the other meaning of the word: "regarded with deep affection". One might engage with ones lover by saying "hello dear". I like skipjack too, but not that much! :lol:.

aows61 August 28th, 2017 06:52 AM

Quote:

Originally Posted by Joppy (Post 578936)
Sorry, i was actually referring to your use of the word 'dear'. Typically we use it when writing a formal letter. For example, "Dear aows61, how are you today?".

However use of 'dear' in the context here ('hello dear') refers to the other meaning of the word: "regarded with deep affection". One might engage with ones lover by saying "hello dear". I like skipjack too, but not that much! :lol:.

thanks for notifying and sharing this info with us,
if you have more info regarding the question, kindly share it with us...

regards,

Country Boy August 28th, 2017 06:56 AM

Quote:

Originally Posted by aows61 (Post 578886)
Hello dear,

I have a question in complex Fourier series:
Q:/ how do I know whether the function is even or odd? if after solving:
a$_0$ = 2$\pi$
a$_n$ = 0
b$_n$ = -2/$\pi$
C$_n$ = j/n
C$_{-n}$ = -j/n
Can you help me with that?

You seem confused as to what a "Fourier series" is. The "$\displaystyle a_n, b_n$ and "$\displaystyle C_n, C_{-n}$" are different notations. We can write a Fourier series as $\displaystyle a_0+ a_1 \cos(x)+ b_1 \sin(x)+ a_2\cos(2x)+ b_2\sin(2x)+ \cdot\cdot\cdot$
The "$\displaystyle a_n$" terms are the coefficients of cos(nx) and the "$\displaystyle b_n$" terms are coefficients of sin(nx).

Since cos(nx) is an even function, for all n, and sin(nx) is an odd function, for all n, a Fourier series is "odd" if and only if "$\displaystyle b_n= 0$" for all n and "even" if and only if "$\displaystyle a_n= 0$" for all n.

We can also write a Fourier series as $\displaystyle \sum_{n=-\infty}^\infty C_ne^{inx}$. Since $\displaystyle e^{inx}= \cos(nx)+ j \sin(nx)$, $\displaystyle \cos(-nx)= \cos(nx)$ and $\displaystyle \sin(-nx)= -\sin(nx)$, that reduces to the previous case.

skipjack August 28th, 2017 07:22 AM

Isn't that the wrong way round?

aows61 August 28th, 2017 08:33 AM

thanks
 
Quote:

Originally Posted by Country Boy (Post 578949)
You seem confused as to what a "Fourier series" is. The "$\displaystyle a_n, b_n$ and "$\displaystyle C_n, C_{-n}$" are different notations. We can write a Fourier series as $\displaystyle a_0+ a_1 \cos(x)+ b_1 \sin(x)+ a_2\cos(2x)+ b_2\sin(2x)+ \cdot\cdot\cdot$
The "$\displaystyle a_n$" terms are the coefficients of cos(nx) and the "$\displaystyle b_n$" terms are coefficients of sin(nx).

Since cos(nx) is an even function, for all n, and sin(nx) is an odd function, for all n, a Fourier series is "odd" if and only if "$\displaystyle b_n= 0$" for all n and "even" if and only if "$\displaystyle a_n= 0$" for all n.

We can also write a Fourier series as $\displaystyle \sum_{n=-\infty}^\infty C_ne^{inx}$. Since $\displaystyle e^{inx}= \cos(nx)+ j \sin(nx)$, $\displaystyle \cos(-nx)= \cos(nx)$ and $\displaystyle \sin(-nx)= -\sin(nx)$, that reduces to the previous case.

I think you mean odd if a_n = 0.

and my function is this function:

$$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$
with the following given data:

$$ C_o=\pi $$
$$ \frac {ao} 2 = \pi $$
$$ C_n=\frac j n $$
$$C_{-n}= \frac {-j} n $$
$$ a_n=0 $$
$$ b_n=\frac {-2} n $$

and the solutions said it is neither odd nor even and I want to know why it is neither odd nor even .....


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