How do you find the log of a complex number?
Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable z=reiΘ z = r e i Θ is defined by the formula logz=lnr+i(Θ+2nπ)(n∈Z)..
How is log expressed?
logarithm, the exponent or power to which a base must be raised to yield a given number.
Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n.
For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8..
What is log method?
log() method returns the natural logarithm (base e) of a double value as a parameter.
There are various cases : If the argument is NaN or less than zero, then the result is NaN.
If the argument is positive infinity, then the result is positive infinity..
What is the difference between log and ln in complex analysis?
The difference between log and ln is that log is defined for base 10 and ln is denoted for base e.
For example, log of base 2 is represented as log2 and log of base e, i.e. loge = ln (natural log)..
Why is log so important?
Logarithmic functions are important largely because of their relationship to exponential functions.
Logarithms can be used to solve exponential equations and to explore the properties of exponential functions..
- Answer: The function Log(z) is analytic except when z is a negative real number or 0.
So Log(1/(z+i)) is analytic except when z=-i (since then 1/(z+i) is undefined) or when 1/(z+i) is a negative real. - Log(z) is analytic on C∖(−∞,0].
To make the function analytic you have to remove all non-positive real numbers from the complex plane.
To see where Log(z+1) ia anlytic you simply have to choose z such that z+1∉(−∞,0] which means z∉(−∞,−1].
So the answer is C∖(−∞,−1].Mar 15, 2018 - The complex logarithm has some multi-valued issues because the complex exponential has periodic behavior.
That is, ex+2πi=ex, so the logarithm needs to reflect this complication somehow.
For the complex logarithm, consider the fact that i4=1.
Then ln(z)=ln(z⋅i4k)=ln(z)+4kln(i)=ln(z)+2kπi. - Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable z=reiΘ z = r e i Θ is defined by the formula logz=lnr+i(Θ+2nπ)(n∈Z).
