The Fourier transform $\mathit{F}\left(\mathit{w}\right)$ of $\mathit{f}=\mathit{f}\left(\mathit{t}\right)$ is

where $\mathit{c}$ and $\mathit{s}$ are parameters with the default values 1 and –1, respectively. Other common values for $\mathit{c}$ are $1/\left(2\pi \right)$ and $1/\sqrt{2\pi }$, and other common values for $\mathit{s}$ are $1$, $-2\pi$, and $2\pi$.

Find the Fourier transform of sin(t) with default c and s parameters.

syms t w
F = fourier(sin(t),t,w)

F = $$-\pi \hspace{0.17em}\left(\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(w-1\right)-\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(w+1\right)\right)\hspace{0.17em}\mathrm{i}$$

Find the same Fourier transform with $\mathit{c}=1/\left(2\pi \right)$ and $\mathit{s}=1$. Set the parameter values by using the 'FourierParameters' preference. Represent $\pi$ exactly by using sym. Specify the values of c and s as the vector [1/(2*sym(pi)) 1]. Store the previous values returned by sympref so that you can restore them later.

oldVal = sympref('FourierParameters',[1/(2*sym(pi)) 1])

oldVal = $$\left(\begin{array}{cc}1& -1\end{array}\right)$$

F = fourier(sin(t),t,w)

F = 
$$\frac{\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(w-1\right)\hspace{0.17em}\mathrm{i}}{2}-\frac{\delta \text{<a href="matlab:doc symbolic/dirac">dirac</a>}\left(w+1\right)\hspace{0.17em}\mathrm{i}}{2}$$

The preferences you set using sympref persist through your current and future MATLAB sessions. Restore the previous values of c and s to oldVal.

sympref('FourierParameters',oldVal);

Alternatively, you can restore the default values of c and s by specifying the 'default' option.

sympref('FourierParameters','default');

In Symbolic Math Toolbox™, the default value of the Heaviside function at the origin is 1/2. Return the value of heaviside(0). Find the Z-transform of heaviside(x) for this default value of heaviside(0).

syms x
H = heaviside(sym(0))

H = 
$$\frac{1}{2}$$

Z = ztrans(heaviside(x))

Z = 
$$\frac{1}{z-1}+\frac{1}{2}$$

Other common values for the Heaviside function at the origin are 0 and 1. Set heaviside(0) to 1 using the 'HeavisideAtOrigin' preference. Store the previous value returned by sympref so that you can restore it later.

oldVal = sympref('HeavisideAtOrigin',1)

oldVal = 
$$\frac{1}{2}$$

Check if the new value of heaviside(0) is 1. Find the Z-transform of heaviside(x) for this value.

H = heaviside(sym(0))

H = $$1$$

Z = ztrans(heaviside(x))

Z = 
$$\frac{1}{z-1}+1$$

The new output of heaviside(0) modifies the output of ztrans.

The preferences you set using sympref persist through your current and future MATLAB sessions. Restore the previous value of heaviside(0) to oldVal.

sympref('HeavisideAtOrigin',oldVal);

Alternatively, you can restore the default value of 'HeavisideAtOrigin' by specifying the 'default' option.

sympref('HeavisideAtOrigin','default');

By default, symbolic expressions in live scripts are displayed in abbreviated output format, and typeset in mathematical notation. You can turn off abbreviated output format and typesetting using symbolic preferences.

Create a symbolic expression and return the output, which is abbreviated by default.

syms a b c d x 
f = a*x^3 + b*x^2 + c*x + d;
outputAbbrev = sin(f) + cos(f) + tan(f) + log(f) + 1/f

outputAbbrev = 
$$\begin{array}{l}\cos \left({\sigma }_1\right)+\log \left({\sigma }_1\right)+\sin \left({\sigma }_1\right)+\tan \left({\sigma }_1\right)+\frac{1}{{\sigma }_1}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=a\hspace{0.17em}{x}^3+b\hspace{0.17em}{x}^2+c\hspace{0.17em}x+d\end{array}$$

Turn off abbreviated output format by setting the 'AbbreviateOutput' preference to false. Redisplay the expression.

sympref('AbbreviateOutput',false);
outputLong = sin(f) + cos(f) + tan(f) + log(f) + 1/f

outputLong = 
$$\cos \left(a\hspace{0.17em}{x}^3+b\hspace{0.17em}{x}^2+c\hspace{0.17em}x+d\right)+\log \left(a\hspace{0.17em}{x}^3+b\hspace{0.17em}{x}^2+c\hspace{0.17em}x+d\right)+\sin \left(a\hspace{0.17em}{x}^3+b\hspace{0.17em}{x}^2+c\hspace{0.17em}x+d\right)+\tan \left(a\hspace{0.17em}{x}^3+b\hspace{0.17em}{x}^2+c\hspace{0.17em}x+d\right)+\frac{1}{a\hspace{0.17em}{x}^3+b\hspace{0.17em}{x}^2+c\hspace{0.17em}x+d}$$

Create another symbolic expression and return the output, which is typeset in mathematical notation by default. Turn off rendered output and use ASCII output instead by setting the 'TypesetOutput' preference to false. First, show the typeset output.

syms a b c d x 
f = exp(a^b)+pi

f = $$\pi +{\mathrm{e}}^{a^b}$$

Turn off typesetting by setting the 'TypesetOutput' preference to false. Redisplay the expression.

sympref('TypesetOutput',false);
f = exp(a^b)+pi

 
f =
 
pi + exp(a^b)
 

The preferences you set using sympref persist through your current and future MATLAB sessions. Restore the default values of 'AbbreviateOutput' and 'TypesetOutput' by specifying the 'default' option.

sympref('AbbreviateOutput','default');
sympref('TypesetOutput','default');

Display symbolic results in floating-point output format, that is the short, fixed-decimal format with 4 digits after the decimal point.

Create a quadratic equation.

syms x
eq = x^2 - 2e3/sym(pi)*x + 0.5 == 0

eq = 
$$x^2-\frac{2000\hspace{0.17em}x}{\pi }+\frac{1}{2}=0$$

Find the solutions of the equation using solve.

sols = solve(eq,x)

sols = 
$$\left(\begin{array}{c}-\frac{\sqrt{2}\hspace{0.17em}\sqrt{2000000-{\pi }^2}-2000}{2\hspace{0.17em}\pi }\\ \frac{\sqrt{2}\hspace{0.17em}\sqrt{2000000-{\pi }^2}+2000}{2\hspace{0.17em}\pi }\end{array}\right)$$

Set the 'FloatingPointOutput' preference to true. Display the quadratic equation and its solutions in floating-point format.

sympref('FloatingPointOutput',true);
eq

eq = $$x^2-636.6198\hspace{0.17em}x+0.5000=0$$

sols

sols = 
$$\left(\begin{array}{c}7.8540e-04\\ 636.6190\end{array}\right)$$

The floating-point format displays each symbolic number in the short, fixed-decimal format with 4 digits after the decimal point. Setting the 'FloatingPointOutput' preference does not affect the floating-point precision in a symbolic computation. To compute symbolic numbers using floating-point arithmetic, use the vpa function.

Now restore the default value of 'FloatingPointOutput' by specifying the 'default' option. Compute the floating-point approximation of the solutions in 8 significant digits using vpa.

sympref('FloatingPointOutput','default');
sols = vpa(sols,8)

sols = 
$$\left(\begin{array}{c}0.00078539913\\ 636.61899\end{array}\right)$$

Create a symbolic polynomial expression consisting of multiple variables. Display the polynomial in the default order.

syms x y a b
p1 = b^2*x^2 + a^2*x + y^3 + 2

p1 = $$a^2\hspace{0.17em}x+b^2\hspace{0.17em}{x}^2+y^3+2$$

The default option sorts the output in alphabetical order, without distinguishing the different symbolic variables in each monomial term.

Now display the same polynomial in ascending order by setting the preference 'PolynomialDisplayStyle' to 'ascend'.

sympref('PolynomialDisplayStyle','ascend');
p1

p1 = $$2+y^3+a^2\hspace{0.17em}x+b^2\hspace{0.17em}{x}^2$$

The 'ascend' option sorts the output in ascending order based on the importance of the variables. Here, the most important variable x with the highest order in a monomial term is displayed last.

Display the polynomial in descending order by setting the 'PolynomialDisplayStyle' preference to 'descend'.

sympref('PolynomialDisplayStyle','descend');
p1

p1 = $$b^2\hspace{0.17em}{x}^2+a^2\hspace{0.17em}x+y^3+2$$

The preferences you set using sympref persist through your current and future MATLAB sessions. Restore the default value of 'PolynomialDisplayStyle' by specifying the 'default' option.

sympref('PolynomialDisplayStyle','default');

By default, a symbolic matrix in live scripts is set in parentheses (round brackets). You can specify the use of square brackets instead by using sympref.

Create a symbolic matrix consisting of symbolic variables and numbers.

syms x y
A = [x*y, 2; 4, y^2]

A = 
$$\left(\begin{array}{cc}x\hspace{0.17em}y& 2\\ 4& y^2\end{array}\right)$$

Display the matrix with square brackets by setting the 'MatrixWithSquareBrackets' preference to true.

sympref('MatrixWithSquareBrackets',true);
A

A = 
$$\left[\begin{array}{cc}x\hspace{0.17em}y& 2\\ 4& y^2\end{array}\right]$$

The preferences you set using sympref persist through your current and future MATLAB sessions. Restore the default value by specifying the 'default' option.

sympref('MatrixWithSquareBrackets','default');

Instead of saving and restoring individual preferences one by one, you can use sympref to save and restore all symbolic preferences simultaneously.

Return a structure containing the values of all symbolic preferences by using sympref().

oldPrefs = sympref()

oldPrefs = struct with fields:
           FourierParameters: [1    -1]
           HeavisideAtOrigin: 1/2
            AbbreviateOutput: 1
               TypesetOutput: 1
         FloatingPointOutput: 0
      PolynomialDisplayStyle: 'default'
    MatrixWithSquareBrackets: 0

Access the value of each symbolic preference by addressing the field of the structure. Alternatively, you can use the command sympref(pref).

val1 = oldPrefs.FourierParameters

val1 = $$\left(\begin{array}{cc}1& -1\end{array}\right)$$

val2 = oldPrefs.HeavisideAtOrigin

val2 = 
$$\frac{1}{2}$$

val3 = sympref('FourierParameters')

val3 = $$\left(\begin{array}{cc}1& -1\end{array}\right)$$

To modify multiple symbolic preferences simultaneously, you can create a structure prefs that contains the preference values. Use the command sympref(prefs) to set multiple preferences.

prefs.FourierParameters = [1/(2*sym(pi)) 1]

prefs = struct with fields:
    FourierParameters: [1/(2*pi)    1]

prefs.HeavisideAtOrigin = 1

prefs = struct with fields:
    FourierParameters: [1/(2*pi)    1]
    HeavisideAtOrigin: 1

sympref(prefs);

Because symbolic preferences persist through your current and future MATLAB sessions, you need to restore your previous preferences. Restore the saved preferences using sympref(oldPrefs).

sympref(oldPrefs);

Alternatively, you can set all symbolic preferences to their default values by specifying the 'default' option.

sympref('default');