Singularity of sin(√z)/√z at z = 0

[mkbh_10]

Homework Statement

Locate & name the singularity of the function sin(sqrtZ)/Sqrt(Z) ?

Homework Equations

The Attempt at a Solution

At z= 0 i gives 0/0 form so should i apply L hospital's rule & then proceed ?

 

[Hootenanny]

The is no need to consider the fraction as an entire entity, instead, one can separately calculate the order of the numerator and denominator independently and then combine them to find the order of the quotient.

Hence, start by determining the order of the numerator and denominator separately.

 

[mkbh_10]

determining the order of the numerator and denominator ?

 

[Hootenanny]

One can determine the order of a function, at a point, by finding the order of the derivative which is non-vanishing at that point. For example, the function,

$$f(x) = x^2$$

Has order 2 at x=0 since,

$$f(0)=0 \;\;,\;\;f^\prime(0) = 0 \;\;,\;\;f^{\prime\prime}(0)=2\neq0$$

Do you follow?

 

[mkbh_10]

The above function has Order =1 at z= 0 , then ?

 

[Hootenanny]

The above function has Order =1 at z= 0 , then ?

Correct. So, if a function has a singularity of order one what type of singularity is it?

 

[mkbh_10]

i dn't know

 

[Hootenanny]

i dn't know

A function with a positive order, at a given point, means that the Laurent series of the function at that point has no principle part, which means the singularity is ________.