FunctionAdvisor/special_values
return the special values of a given mathematical function
Calling Sequence
Parameters
Description
Examples
FunctionAdvisor(special_values, math_function)
special_values
-
literal name; 'special_values'
math_function
Maple name of mathematical function
The FunctionAdvisor(special_values, math_function) command returns special values of the function computed at a selected list of special points.
FunctionAdvisorspecial_values,arccsc
arccsc−1=−π2,arccsc−233=−π3,arccsc−2=−π4,arccsc−2=−π6,arccsc2=π6,arccsc2=π4,arccsc233=π3,arccsc1=π2,arccscI=−Iln1+2,arccsc−I=Iln1+2,arccsc0=π2−∞I,arccsc∞=0,arccsc−∞=0
FunctionAdvisorspecial_values,sin
sinπ6=12,sinπ4=22,sinπ3=32,sin∞=undefined,sin∞I=∞I,sinπn=0,n::ℤ,sin2n+1π2=−1,n::odd,sin2n+1π2=1,n::even
ex1≔FunctionAdvisorspecial_values,binomial
ex1≔0z=1z=0sinπzπzotherwise,a0=1,a1=a,00=1
The variables used by the FunctionAdvisor command to create the function calling sequences are local variables. Therefore, the previous example does not depend on a or z.
dependsex1,a,dependsex1,z
false,false
To make the FunctionAdvisor command return results using global variables, pass the function call itself.
ex2≔FunctionAdvisorspecial_values,binomiala,z
ex2≔0z=1z=0sinπzπzotherwise,a0=1,a1=a,00=1
dependsex2,a,dependsex2,z
true,true
For functions which accept more than one argument, the special values of interest could be restricted by passing the function call. For example, these are special values for Ψ1,z
FunctionAdvisorspecial_values,Ψ1,z
Ψ−1,z=lnGAMMAz−ln2π2,Ψ1,1=π26,Ψ1,12=π22,Ψ1,14=π2+8Catalan,Ψ0,z=Ψz,Ψ1,−1=∞,Ψ1,0=∞,Ψ1,−n=∞,Ψ1,n=π26−∑k=1n−11k2,Ψ1,n+12=π22−∑k=1n42k−12,Ψ1,12−n=π22+∑k=1n42k−12,n::ℤ+
and these are special values for Ψz.
FunctionAdvisorspecial_values,Ψz
Ψ−32=83−γ−2ln2,Ψ−12=2−γ−2ln2,Ψ14=−γ−3ln2−π2,Ψ12=−γ−2ln2,Ψ34=−γ−3ln2+π2,Ψ13=−γ−π36−3ln32,Ψ23=−γ+π36−3ln32,Ψ1=−γ,Ψ2=1−γ,Ψ−1=∞+∞I,Ψ0=∞+∞I,Ψ∞=∞,Ψ−∞=undefined,Ψ∞I=∞+Iπ2,Ψ−∞I=∞−Iπ2,Ψn=∑k=1n−11k−γ,n::ℤ+,Ψn+pq=q∑k=0n−11kq+p+2∑k=1q2−12cos2πpkqlnsinπkq−πcotπpq2−ln2q−γ,Ψpq−n=q∑k=0n−11qk+1−p+2∑k=1q2−12cos2πpkqlnsinπkq−πcotπpq2−ln2q−γ,n::ℤ0,+∧p,q::setℤ+∧p<q
In these cases, when the FunctionAdvisor command is called with the function name, for example, Psi, the values are listed on the screen starting with those involving less arguments in the function call (in this example, Ψz).
See Also
depends
FunctionAdvisor
FunctionAdvisor/ branch_cuts
FunctionAdvisor/ branch_points
FunctionAdvisor/ identities
FunctionAdvisor/ topics
Psi
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