Difference between revisions of "Logarithm (multivalued)"
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(Created page with "The (multivalued) logarithm function $\mathrm{Log} \colon \mathbb{C} \rightarrow \mathscr{P}\left( \mathbb{C} \right)$ is defined by $$\mathrm{Log}(z)=\displaystyle\int_1^z \d...") |
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$$\mathrm{Log}(z)=\displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$ | $$\mathrm{Log}(z)=\displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$ | ||
where $\mathscr{P} \left( \mathbb{C} \right)$ denotes the [[power set]] of $\mathbb{C}$ and where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not cross $0$. | where $\mathscr{P} \left( \mathbb{C} \right)$ denotes the [[power set]] of $\mathbb{C}$ and where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not cross $0$. | ||
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| + | ==Properties== | ||
| + | [[Real and imaginary parts of log]]<br /> | ||
| + | [[Logarithm (multivalued) of product is a sum of logarithms (multivalued)]]<br /> | ||
| + | [[Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued)]]<br /> | ||
| + | [[Relationship between logarithm (multivalued) and positive integer exponents]]<br /> | ||
==References== | ==References== | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polar coordinates|next=Relationship between logarithm (multivalued) and logarithm}}: 4.1.4 | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Polar coordinates|next=Relationship between logarithm (multivalued) and logarithm}}: 4.1.4 | ||
Revision as of 06:55, 4 June 2016
The (multivalued) logarithm function $\mathrm{Log} \colon \mathbb{C} \rightarrow \mathscr{P}\left( \mathbb{C} \right)$ is defined by $$\mathrm{Log}(z)=\displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$ where $\mathscr{P} \left( \mathbb{C} \right)$ denotes the power set of $\mathbb{C}$ and where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not cross $0$.
Properties
Real and imaginary parts of log
Logarithm (multivalued) of product is a sum of logarithms (multivalued)
Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued)
Relationship between logarithm (multivalued) and positive integer exponents
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.4
