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Logarithmic derivative

Given a function , the ratio

of its first derivative and itself is called the logarithmic derivative of .

When is a real function of real variable and takes strictly positive values then the chain rules gives

The observation is the motivation for the name.  

The logarithmic derivative has many useful properties, For instance

• The logarithmic derivative of the product of functions is the sum of their logarithmic derivatives   
• The logarithmic derivative of the quotient of functions is the difference of their logarithmic derivatives  
• The logarithmic derivative of the th power of a function is   times the logarithmic derivative of the function  
• The logarithmic derivative of the exponential of a function equals the derivative of a function  .

In the theory of complex functions we have:

Theorem. Let be a meromorphic function in the region (i.e. open and connected set) and be a region such that its closure and which boundary is a continuous curve not containing a zero o pole of . If   and denotes the number of zeros and poles of lying inside , respectively, then

where is an oriented boundary.

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