# integration property of fourier series

The integration property in Fourier series is as follows:

So, for proving the above property, i took this approach:

This is where my doubt is. Some books and websites just put upper limit (ignoring the lower limit) and compare with (1) to conclude that

However, when the lower integral limit is substituted, we get

This value cannot be evaluated. Though some websites say that this corresponds to the "initial value"(?) and that is zero... etc., how does one justify this mathematically?

Any help is appreciated. Thank you.

• just a friendly suggestion: in the future, please learn to use $\LaTeX$ and express your equations with source code rather than screenshots. i want to just be able to copy them without re-expression (in $\LaTeX$). make it easier for people to help you rather than harder for them to help. Commented Jun 17, 2019 at 8:14
• Hil, using a reverse argument (what would the resulting Fourier series have to be if you differentiated it?) you could come up with all of the Fourier coefficients except for the DC coefficient. That would be undetermined unless the bottom limit of the integral was known and set to a finite value. Commented Jun 17, 2019 at 8:22
• BTW, the notation for a definite integral from whatever the book that was screen shot is horrible: $$\int\limits_{-\infty}^{t} x(t) \, dt$$ the dummy variable of integration should not be the same as the upper limit of the integral in an definite integral. this notation is legit: $$\int\limits_{-\infty}^{t} x(u) \, du$$ Commented Jun 17, 2019 at 8:26
• so Nish, have you seen how to render equations with $\LaTeX$? Commented Jun 17, 2019 at 18:41

Note that the antiderivative of a function is only defined up to a constant. Furthermore, note that if you integrate a periodic function, the result is not necessarily periodic. Let

$$x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t}\tag{1}$$

Now we integrate $$(1)$$ with an arbitrary lower integration limit $$t_0$$. I'll say more about that later.

\begin{align}y(t)&=\int_{t_0}^tx(\tau)d\tau\\&=\sum_{k=-\infty}^{\infty}a_k\int_{t_0}^te^{jk\omega_0\tau}d\tau\\&=a_0(t-t_0)+\sum_{k=-\infty\\k\neq 0}^{\infty}\frac{a_k}{jk\omega_0}e^{jk\omega_0t}-\sum_{k=-\infty\\k\neq 0}^{\infty}\frac{a_k}{jk\omega_0}e^{jk\omega_0t_0}\tag{2}\end{align}

Clearly, if $$a_0\neq 0$$, $$y(t)$$ isn't periodic, so we can't hope to obtain a formula for its Fourier coefficients. So in order to end up with a periodic function $$y(t)$$, we require that $$a_0=0$$.

If $$a_0=0$$, we get from $$(2)$$

$$y(t)=\sum_{k=-\infty}^{\infty}b_ke^{jk\omega_0t}\tag{3}$$

with

$$b_k=\begin{cases}\displaystyle\frac{a_k}{jk\omega_0},&k\neq 0\\\displaystyle-\sum_{l=-\infty\\l\neq 0}^{\infty}\frac{a_l}{jl\omega_0}e^{jl\omega_0t_0},&k=0\end{cases}\tag{4}$$

So in general the antiderivative $$y(t)$$ has a non-zero DC component, which depends on the choice of the lower integration limit $$t_0$$.

The antiderivative

$$\tilde{y}(t)=\sum_{k=-\infty\\k\neq 0}^{\infty}\frac{a_k}{jk\omega_0}e^{jk\omega_0t}\tag{5}$$

is the specific antiderivative of $$x(t)$$ that has a zero DC Fourier coefficient, and this is the one that is meant by the formula that is usually given in textbooks.

Consequently, the expression

$$y(t)=\int_{-\infty}^tx(\tau)d\tau\tag{6}$$

for the antiderivative of $$x(t)$$ with Fourier coefficients $$b_k=a_k/(jk\omega_0)$$, $$k\neq 0$$, and $$b_0=0$$, is at least inaccurate; it is in fact a sloppy way of expressing the specific antiderivative $$\tilde{y}(t)$$ given by $$(5)$$, i.e., the one with a zero DC coefficient.

• i like Eq. (4). Commented Jun 17, 2019 at 19:51
• @MattL. Thank you very much for your insight sir. However, this doesn't answer one of the queries i have asked in the post i.e, what happens when $$t_{o} = \infty ?$$ The complex exponential in equation 4 still doesn't converge to any value right? Commented Jul 6, 2019 at 16:09
• @NishanthARao: As you've already noticed, that limit doesn't exist. Commented Jul 6, 2019 at 18:19
• @MattL. I am very grateful for your detailed answer. However with due respect, my main question was what happens to the complex exponential at $t_{o} = - \infty$ ? In both your approach and my approach, we dont come to an exact conclusion on what happens when $t_{o} \to \infty$ or in my case, $e^{-jkw_{o} \infty}$. I just wanted to know how to get the answer from our approach, and conceptually where is it going wrong. I am, totally thankful for robertbristow-johnson and you, for clearing my doubt. Commented Jul 23, 2019 at 10:40
• @Jarvis: The answer is given in my previous comment. That limit doesn't exist. Commented Jul 23, 2019 at 10:44

Oh, hell...

Let's say you have two periodic functions, $$x(t)$$ and $$y(t)$$ having exactly the same period (and fundamental frequency):

$$x(t) \triangleq \sum\limits_{k=-\infty}^{\infty} a_k \, e^{j k \omega_0 t}$$

$$y(t) \triangleq \sum\limits_{k=-\infty}^{\infty} b_k \, e^{j k \omega_0 t}$$

where the period common to both is $$\frac{2 \pi}{\omega_0}$$.

Suppose you were to differentiate $$y(t)$$:

$$y'(t) = \sum\limits_{k=-\infty}^{\infty} j k \omega_0 b_k \, e^{j k \omega_0 t}$$

you can see right away that the DC term of $$y'(t)$$ is zero (as it should be):

$$j k \omega_0 b_k \bigg|_{k=0} = 0$$

Now let's say that you set that differentiated periodic function to $$x(t)$$:

$$x(t) = y'(t)$$

That means $$y(t)$$ is the indefinite integral of $$x(t)$$. Then you see that all of the coefficients of $$y(t)$$ are well-defined in terms of the coefficients of $$x(t)$$ except for the DC coefficient, $$b_0$$, which could be any finite value.

$$a_k = j k \omega_0 b_k$$

or

$$b_k = \frac{a_k}{j k \omega_0} \qquad \qquad \forall k \in \mathbb{Z} \ne 0$$.

That is, in my opinion, the only legit way to look at the integration of a Fourier series.

• This is a wonderful way to look at the property. Extremely thankful for your insight sir. With due respect however, I just want to know where I am going wrong, and if at all, why? I am pretty sure there must be some way to continue from that step. Commented Jun 17, 2019 at 8:59
• @NishanthARao: rbj's answer is probably the most straightforward way to look at the problem. If you're interested in how to figure out the solution by integration, have a look at my answer. Commented Jun 17, 2019 at 10:58

It appears to me that the "initial value" issue would affect only the DC value, $$a_0$$, of the Fourier series. In my opinion, the author should have left the integrals as indefinite integrals (which are the same as the "anti-derivative") or have expressed this relationship only in terms of differentiation, and not integration.

Actually, now that I think of it, the original $$x(t)$$ would have to have $$a_0 = 0$$ in order for the integration theorem to make any sense. And the resulting Fourier series can have any finite DC value you want. You would have to determine the resulting DC value by other means.

To do this theorem properly, you must first do it for the derivative of $$x(t)$$ first and then apply the results in reverse.

I dunno what book you're using but there are both some notational and even mathematical problems with the expression of this problem.

• i'm gonna let someone else (maybe @MattL. ) make the appropriate and rigorous answer to this question. as it is posed, it's sorta a flawed problem in definition. (and i do not blame the OP for this, it could be the fault of the book.) Commented Jun 17, 2019 at 8:34

I am sure my answer is way too late. I would rather establish the integration property by using the analysis equation instead.

Also, you should remember that you need the integral of $$x(t)$$ to be periodic. That is, $$a_0 = 0$$.

Call $$X(t)$$ to the $$\int_{-\infty}^t x(\tau) d\tau$$. Clearly, $$X(t+T) = X(t)$$, which is crucial for the proof to kick in.

Now, use the analysis equation for $$k \neq 0$$

$$b_k = 1/T \int_T X(t) e^{-jkw_0t} dt$$

Do primitivation by parts, defining $$u = X(t)$$ and $$v' = e^{-jkw_0t}$$.

The primitive of $$X(t) e^{-jkw_0t}$$, written as $$P(X(t) e^{-jkw_0t})$$, is $$\frac{X(t)}{-jkw_0} e^{-jkw_0t} - P(\frac{x(t)}{-jkw_0} e^{-jkw_0t})$$

The term $$uv = \frac{X(t)}{-jkw_0} e^{-jkw_0t}$$ integrates to zero over any interval of size $$T$$, due to the periodicity of $$X(t)$$ and that of $$e^{-jkw_0t}$$.

Hence, you are solely left with the primitive of $$-u'v$$, with $$-u'v$$ being

$$\frac{x(t)}{jkw_0} e^{-jkw_0t}$$

That is,

$$b_k = 1/T \int_T X(t) e^{-jkw_0t} dt = \frac{1}{T} \frac{1}{jkw_0} \int_T x(t) e^{-jkw_0t} dt = \frac{a_k}{jkw_0}$$

which is the result we wanted to establish.

As for $$b_0$$, it may be non zero, although $$a_0 = 0$$. Its calculation is simply $$b_0 = 1/T \int_T X(t) dt$$, as $$b_0$$ is not obtainable from $$a_0$$.