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Function Repository Resource:

OddFunctionQ

Source Notebook

Determine whether an expression is an odd function of the given variable or variables

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["OddFunctionQ"][expr,x]

returns True if expr is an odd function of x, and returns False otherwise.

ResourceFunction["OddFunctionQ"][expr,{x1,x2,}]

returns True if expr is an odd function under the transformation {x1,x2,}{-x1,-x2,}, and returns False otherwise.

Examples

Basic Examples

Test whether a basic power function is odd:

In[1]:=
ResourceFunction["OddFunctionQ"][x^3, x]
Out[1]=


Test another power function:

In[2]:=
ResourceFunction["OddFunctionQ"][x^2, x]
Out[2]=


Test whether a constant function is odd:

In[3]:=
ResourceFunction["OddFunctionQ"][1, x]
Out[3]=


Test whether the absolute value function is odd:

In[4]:=
ResourceFunction["OddFunctionQ"][Abs[x], x]
Out[4]=


Test whether the sine function is odd:

In[5]:=
ResourceFunction["OddFunctionQ"][Sin[x], x]
Out[5]=


Test whether the cosine function is odd:

In[6]:=
ResourceFunction["OddFunctionQ"][Cos[x], x]
Out[6]=


Test whether a gaussian function is odd:

In[7]:=
ResourceFunction["OddFunctionQ"][Exp[-x^2], x]
Out[7]=


Test a signed gaussian function:

In[8]:=
ResourceFunction["OddFunctionQ"][Sign[x] Exp[-x^2], x]
Out[8]=


Test whether a hyperbolic sine function of two variables is odd:

In[9]:=
ResourceFunction["OddFunctionQ"][Sinh[x + y], {x, y}]
Out[9]=


Test whether the hyperbolic tangent function is odd:

In[10]:=
ResourceFunction["OddFunctionQ"][Tanh[x], x]
Out[10]=


Test whether the error function is odd:

In[11]:=
ResourceFunction["OddFunctionQ"][Erf[x], x]
Out[11]=


Test whether a Fresnel integral is odd:

In[12]:=
ResourceFunction["OddFunctionQ"][FresnelC[x], x]
Out[12]=


Test whether a shifted Fresnel integral is odd:

In[13]:=
ResourceFunction["OddFunctionQ"][FresnelC[x] + 1, x]
Out[13]=

Scope


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:

In[14]:=
ResourceFunction["OddFunctionQ"][Cosh[x + y], {x, y}]
Out[14]=
In[15]:=
ResourceFunction["OddFunctionQ"][Sinh[x + y], {x, y}]
Out[15]=

Find the parity of a function of three variables:

In[16]:=
ResourceFunction["OddFunctionQ"][Sin[x y z], {x, y, z}]
Out[16]=

Wrapping one of the variables in Abs converts this to an even function:

In[17]:=
ResourceFunction["OddFunctionQ"][Sin[x y Abs[z]], {x, y, z}]
Out[17]=

Resource History

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