Basic Examples

In[]:=
Find the parity of a basic power function:
[◼]
FunctionParity
[x^2,x]
In[]:=
1
Out[]=
Test another power function:
[◼]
FunctionParity
[x^3,x]
In[]:=
-1
Out[]=
Show that a constant function has even parity:
[◼]
FunctionParity
[1,x]
In[]:=
1
Out[]=
Find the parity of a rational function:
[◼]
FunctionParity
[x/(x^2+1),x]
In[]:=
-1
Out[]=
Find the parity of the absolute value function:
[◼]
FunctionParity
[Abs[x],x]
In[]:=
1
Out[]=
Find the parity of a trigonometric function:
[◼]
FunctionParity
[Sin[x+Pi/4]+Cos[x+Pi/4],x]
In[]:=
1
Out[]=
Find the parity of a gaussian function:
[◼]
FunctionParity
[Exp[-x^2],x]
In[]:=
1
Out[]=
Test a signed gaussian function:
[◼]
FunctionParity
[Sign[x]Exp[-x^2],x]
In[]:=
-1
Out[]=
Find the parity of the hyperbolic tangent function:
[◼]
FunctionParity
[Tanh[x],x]
In[]:=
-1
Out[]=
Find the parity of the error function:
[◼]
FunctionParity
[Erf[x],x]
In[]:=
-1
Out[]=
Find the parity of a Fresnel integral:
[◼]
FunctionParity
[FresnelC[x],x]
In[]:=
-1
Out[]=
Find the parity of a shifted Fresnel integral:
[◼]
FunctionParity
[FresnelC[x]+1,x]
In[]:=
Undefined
Out[]=

Scope

In[]:=
For a function of multiple variables, parity is determined based on the behavior of the function under the transformation
{x,y,..}{-x,-y,..}
. Find the parity of a function of two variables:
[◼]
FunctionParity
[Sinh[x+y],{x,y}]
In[]:=
-1
Out[]=
Find the parity of a function of three variables:
[◼]
FunctionParity
[Sin[xyz],{x,y,z}]
In[]:=
-1
Out[]=
Wrapping
z
in
Abs
converts this to an even function:
[◼]
FunctionParity
[Sin[xyAbs[z]],{x,y,z}]
In[]:=
1
Out[]=
Show that an implicitly-defined circle is even in both
x
and
y
:
[◼]
FunctionParity
[x^2+y^2,{x,y}]
In[]:=
1
Out[]=

Properties and Relations

In[]:=
FunctionParity returns
Undefined
for functions that are neither even nor odd:
[◼]
FunctionParity
[1+x,x]
In[]:=
Undefined
Out[]=
The constant function
f(x)=0
is both even and odd:
[◼]
FunctionParity
[0,x]
In[]:=
0
Out[]=