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The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral +, the substitution + = + can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
For Lebesgue measurable functions, the theorem can be stated in the following form: [6] Theorem — Let U be a measurable subset of R n and φ : U → R n an injective function , and suppose for every x in U there exists φ ′( x ) in R n , n such that φ ( y ) = φ ( x ) + φ′ ( x )( y − x ) + o (‖ y − x ‖) as y → x (here o is ...
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.
For example, E can be Euclidean n-space R n or some Lebesgue measurable subset of it, X is the σ-algebra of all Lebesgue measurable subsets of E, and μ is the Lebesgue measure. In the mathematical theory of probability, we confine our study to a probability measure μ , which satisfies μ ( E ) = 1 .
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where a ( x ) = a ∈ R {\displaystyle a(x)=a\in \mathbb {R} } is constant, b ( x ) = x , {\displaystyle b(x)=x,} and f ( x , t ...
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution .
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.
Dominated convergence theorem (Lebesgue integration) Egorov's theorem (measure theory) Fatou–Lebesgue theorem (real analysis) Fubini's theorem (integration) Hahn decomposition theorem (measure theory) Hahn–Kolmogorov theorem (measure theory) Ham sandwich theorem ; Hobby–Rice theorem (mathematical analysis) Kōmura's theorem (measure theory)