---
title: "FourierSinCoefficient"
language: "en"
type: "Symbol"
summary: "FourierSinCoefficient[expr, t, n] gives the n\\[Null]^th coefficient in the Fourier sine series expansion of expr. FourierSinCoefficient[expr, {t1, t2, ...}, {n1, n2, ...}] gives a multidimensional Fourier sine coefficient."
keywords: 
- Fourier sine series
- Fourier sine expansion
- odd function Fourier series
- orthogonal function series
- orthogonal function expansion
- Joseph Fourier
- boundary value problem
- Sturm-Liouville problem
- odd function
canonical_url: "https://reference.wolfram.com/language/ref/FourierSinCoefficient.html"
source: "Wolfram Language Documentation"
related_guides: 
  - 
    title: "Fourier Analysis"
    link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md"
  - 
    title: "Integral Transforms"
    link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md"
related_functions: 
  - 
    title: "FourierSinSeries"
    link: "https://reference.wolfram.com/language/ref/FourierSinSeries.en.md"
  - 
    title: "FourierDST"
    link: "https://reference.wolfram.com/language/ref/FourierDST.en.md"
  - 
    title: "FourierCosCoefficient"
    link: "https://reference.wolfram.com/language/ref/FourierCosCoefficient.en.md"
  - 
    title: "FourierCoefficient"
    link: "https://reference.wolfram.com/language/ref/FourierCoefficient.en.md"
  - 
    title: "Fourier"
    link: "https://reference.wolfram.com/language/ref/Fourier.en.md"
  - 
    title: "FourierSinTransform"
    link: "https://reference.wolfram.com/language/ref/FourierSinTransform.en.md"
  - 
    title: "Integrate"
    link: "https://reference.wolfram.com/language/ref/Integrate.en.md"
---
# FourierSinCoefficient

FourierSinCoefficient[expr, t, n] gives the n$$^{\text{th}}$$ coefficient in the Fourier sine series expansion of expr.

FourierSinCoefficient[expr, {t1, t2, …}, {n1, n2, …}] gives a multidimensional Fourier sine coefficient*.*

## Details and Options

* The $n$$$^{\text{th}}$$ coefficient in the Fourier sine series expansion of $f(t)$ is by default given by $\frac{2}{\pi }\int _0^{\pi }f(t) \sin (n t)dt$.

* The $m$-dimensional Fourier sine coefficient is given by $\left(\frac{2}{\pi }\right)^m\int _0^{\pi }\int _0^{\pi }\cdots  f(t) \sin \left(n_1 t_1\right)\sin \left(n_2 t_2\right)\cdots dt_1dt_2\cdots$.

* In the form ``FourierSinCoefficient[expr, t, n]``, ``n`` can be symbolic or a positive integer.

* The following options can be given:

|                    |               |                                                                   |
| ------------------ | ------------- | ----------------------------------------------------------------- |
| Assumptions        | \$Assumptions | assumptions on parameters                                         |
| FourierParameters  | {1, 1}        | parameters to define Fourier series                               |
| GenerateConditions | False         | whether to generate results that involve conditions on parameters |

* The function ``expr`` is assumed to be periodic in ``t`` with period $2 \pi$, except when otherwise specified by ``FourierParameters``.

* Common settings for ``FourierParameters`` include:

|     |     |     |
| --- | --- | --- |
| {1, 1} | $\frac{2 }{\pi }\int _0^{\pi }f(t) \sin (n t)dt$ | default settings |
| {1, 2Pi} | $4\int _0^{\frac{1}{2}}f(t) \sin (2 \pi  n t)dt$ | period 1 |
| {a, b} | $\left\| \frac{2 b}{\pi }\right\| ^{\frac{a+1}{2}}\int _0^{\frac{\pi }{\| b\| }}f(t) \sin (b n t)dt$ | general setting |

## Examples (6)

### Basic Examples (2)

Find the 3$$^{\text{rd}}$$ Fourier sine series coefficient:

```wl
In[1]:= FourierSinCoefficient[Exp[-t], t, 3]

Out[1]= (3 (1 + E^-π)/5 π)
```

Find the general term coefficient:

```wl
In[2]:= FourierSinCoefficient[Exp[-t], t, n]

Out[2]= ((2 - 2 (-1)^n E^-π) n/(1 + n^2) π)
```

Plot the coefficient sequence:

```wl
In[3]:= DiscretePlot[%, {n, -7, 7}]

Out[3]= [image]
```

---

Find the ``{3, 5}`` Fourier sine coefficient:

```wl
In[1]:= FourierSinCoefficient[x y ^ 2, {x, y}, {3, 5}]

Out[1]= (4 (-4 + 25 π^2)/375 π)
```

The general term:

```wl
In[2]:= FourierSinCoefficient[x y ^ 2, {x, y}, {m, n}]

Out[2]= (4 (-1)^m (2 - 2 (-1)^n + (-1)^n n^2 π^2)/m n^3 π)
```

Plot the coefficient sequence:

```wl
In[3]:= ListPointPlot3D[Table[%, {m, -5, 5}, {n, -5, 5}], Filling -> Bottom, DataRange -> {{-5, 5}, {-5, 5}}, PlotRange -> All]//Quiet

Out[3]= [image]
```

### Scope (4)

Find the 5$$^{\text{th}}$$ Fourier sine coefficient for a quadratic polynomial:

```wl
In[1]:= FourierSinCoefficient[t ^ 2 + 5t + 1, t, 5]

Out[1]= 2 + (92/125 π) + (2 π/5)
```

---

Find the general coefficient for a piecewise function:

```wl
In[1]:= FourierSinCoefficient[UnitStep[t(Pi / 2 - t)], t, n]

Out[1]= (4 Sin[(n π/4)]^2/n π)
```

---

General Fourier sine coefficient for a Gaussian function:

```wl
In[1]:= FourierSinCoefficient[E ^ (-x ^ 2), x, n]

Out[1]= (E^-(n^2/4) (2 Erfi[(n/2)] - Erfi[(n/2) - I π] - Erfi[(n/2) + I π])/2 Sqrt[π])
```

---

Fourier sine coefficient for a basis function:

```wl
In[1]:= FourierSinCoefficient[Sin[3t], t, n]

Out[1]= DiscreteDelta[-3 + n]

In[2]:= Table[%, {n, 1, 10}]

Out[2]= {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}
```

## See Also

* [`FourierSinSeries`](https://reference.wolfram.com/language/ref/FourierSinSeries.en.md)
* [`FourierDST`](https://reference.wolfram.com/language/ref/FourierDST.en.md)
* [`FourierCosCoefficient`](https://reference.wolfram.com/language/ref/FourierCosCoefficient.en.md)
* [`FourierCoefficient`](https://reference.wolfram.com/language/ref/FourierCoefficient.en.md)
* [`Fourier`](https://reference.wolfram.com/language/ref/Fourier.en.md)
* [`FourierSinTransform`](https://reference.wolfram.com/language/ref/FourierSinTransform.en.md)
* [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md)

## Related Guides

* [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md)
* [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md)

## History

* [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md)