Theorem of raising

The theorem of raising is employed at the time of the study of real functions to complex values. It can be stated in the following way:

If $f$ is a continuous function on an interval $I$ of $\backslash \; mathbb\; R$ at values in the circle unit $\backslash \; mathbb\; U$ of the Plan complex, then there exists a continuous application $\backslash \; theta$ on $I$ to values in $\backslash \; mathbb\; R$ such as:

$\backslash \; forall\; \{T\}\; \backslash \; in\; \{I\},\; \backslash \; \{F\; (T)\; =\; \backslash \; exp\; (I\; \backslash \; theta\; (T))\}$

It is said whereas $\backslash \; theta$ is a raising of $f$.

Complements

Two raisings of $f$ differ from a constant of the form $2\; \{K\}\; \{\backslash \; pi\}$ where $k$ is a relative entirety.

If the function $f$ is of class $C^k$ with $k$ a natural entirety, then there exists a raising $\backslash \; theta$ of $f$ of class $C^k$.

Methods of demonstration

For $k\; >\; 0$ the demonstration consists in deriving the relation then to define $\backslash \; theta$ by an integral.

For $k\; =\; 0$ the demonstration appreciably different and is called upon various concepts of topology, in particular the uniform Continuité.

See too

• Parameter setting

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