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FunctionAdvisor/branch_points

return the branch points of a given mathematical function

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

FunctionAdvisor(branch_points, math_function)

Parameters

branch_points

-

literal name; 'branch_points'

math_function

-

Maple name of mathematical function

Description

• 

The FunctionAdvisor(branch_points, math_function) command returns the branch points of the function, if any, or the string "No branch points". If the requested information is not available, the FunctionAdvisor command returns NULL.

• 

A branch point of an analytic function is a point in the complex plane which can be mapped to different values of the argument of the function. Also, the function is not analytic in the neighborhood of a branch point, so these points are not isolated singularities.

  

For example, the origin is a branch point for the square root function because this function will change sign along any path that encircles the origin.  To see this, write the square root function in polar notation z=r&ExponentialE;I2θ where r&equals;z and the angle θ lies in 0θ<2π . At θ=0, z&equals;r. As θ approaches 2π, the same point is approached, but now, zr&ExponentialE;πI=r .  Note that this example does not define the branch cut of the square root function as the positive real axis.  The branch cut of the square root function is the negative real axis because the range of the argument function is π,π .

• 

In general, branch points are related to the presence of branch cuts and the related multivaluedness of the function in the complex plane. Branch points frequently signal the endpoints of a branch cut line or segment.

  

For the isolated singularities of a mathematical function, see singularities. For computing the singular points of an expression, see singular.

Examples

FunctionAdvisorbranch_points&comma;arcsin

arcsinz&comma;z&in;1&comma;1&comma;&infin;&plus;&infin;I

(1)

FunctionAdvisorbranch_cuts&comma;arcsin

arcsinz&comma;z1&comma;1z

(2)

FunctionAdvisorbranch_points&comma;BesselK

BesselKa&comma;z&comma;z&in;0&comma;&infin;&plus;&infin;I

(3)

FunctionAdvisorbranch_points&comma;exp

&ExponentialE;z&comma;No branch points

(4)

See Also

FunctionAdvisor

FunctionAdvisor/branch_cuts

FunctionAdvisor/singularities

FunctionAdvisor/topics

singular

 


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