Show that for real $x$, $$\varphi _ { n } ( x ) = \sum _ { k = 0 } ^ { + \infty } \frac { x ^ { k } } { k ! } I _ { n , k } \quad \text { with } \quad I _ { n , k } = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } i ^ { k } e ^ { - i n t } ( \sin t ) ^ { k } \mathrm {~d} t$$
Using Euler's formula, justify that for $( n , k )$ in $\mathbb { N } \times \mathbb { N }$, $$I _ { n , k } = \sum _ { m = 0 } ^ { k } \frac { A _ { m , k } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { i t ( 2 m - k - n ) } \mathrm { d } t$$ with $A _ { m , k }$ constants to be determined.
Verify that $$\begin{cases} I _ { n , k } = 0 & \text { if } n > k \text { or if } k - n \text { is odd } \\ I _ { n , k } = \frac { ( - 1 ) ^ { p } } { 2 ^ { n + 2 p } } \binom { n + 2 p } { n + p } & \text { if } k = n + 2 p \text { with } p \geqslant 0 \end{cases}$$
Deduce the power series development, for $n \geqslant 0$ and $x \in \mathbb { R }$ : $$\varphi _ { n } ( x ) = \sum _ { p = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { p } } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$ Specify the radius of convergence.
Let $n$ in $\mathbb { N } ^ { * }$, verify that for real $x$ $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( x ^ { n } \varphi _ { n } ( x ) \right) = x ^ { n } \varphi _ { n - 1 } ( x )$$
We approximate $\varphi _ { n } ( x )$ using partial sums $$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$ From which value $p _ { 0 }$ of $p$ is the sequence $\left( a _ { p } \right) _ { p \in \mathbb { N } }$ decreasing?
We approximate $\varphi _ { n } ( x )$ using partial sums $$S _ { m } = \sum _ { p = 0 } ^ { m } ( - 1 ) ^ { p } a _ { p } \quad \text { with } \quad m \in \mathbb { N } \quad \text { and } \quad a _ { p } = \frac { 1 } { p ! ( n + p ) ! } \left( \frac { x } { 2 } \right) ^ { n + 2 p }$$ Assume $N > p _ { 0 }$. Bound $\left| R _ { N } \right|$ as a function of $(N, n, x)$ with $$R _ { N } = \sum _ { p = N + 1 } ^ { + \infty } ( - 1 ) ^ { p } a _ { p }$$ Deduce, for fixed $\varepsilon > 0$, a sufficient condition on $N$ for $\left| \varphi _ { n } ( x ) - S _ { N } \right| < \varepsilon$.
We study the differential equation with unknown $y$ $$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$ We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$. Using the power series development of $\varphi _ { n }$ (II.C.3), show that $\varphi _ { n }$ is a solution on $[ 0 , + \infty [$ of (III.1).
We study the differential equation with unknown $y$ $$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$ We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$. Let $y$ be a solution in $E$ of (III.1). We set $z ( x ) = \sqrt { x } y ( x )$ for all $x \in ] 0 , + \infty [$. Show that $z$ is a solution in $E$ of a differential equation of the type $$z ^ { \prime \prime } + q z = 0 \tag{III.2}$$ with $q \in E$. Specify the expression of the function $q$ and verify that $\lim _ { x \rightarrow + \infty } q ( x ) = 1$.
We study the differential equation with unknown $y$ $$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$ We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$. Let $z$ be a solution of $z^{\prime\prime} + qz = 0$ (III.2). Justify that if $z$ is a non-zero solution of (III.2), then for $x > 0 , \left( z ( x ) , z ^ { \prime } ( x ) \right) \neq ( 0,0 )$. Deduce that if $\alpha$ is a zero of $z$, then there exists a strictly positive real $\eta$ such that $\alpha$ is the only point where $z$ vanishes on $I = ] \alpha - \eta , \alpha + \eta [$. In this case, we say that $\alpha$ is an isolated zero of $z$.
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ : $$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$ Let $x _ { 0 }$ in $] 0 , + \infty [$. By considering the differential equation (III.3) in the form $z ^ { \prime \prime } + z = g$ with $g ( x ) = \frac { - \lambda } { x ^ { 2 } } z ( x )$, solve it on $] 0 , + \infty [$ using the method of variation of constants. Deduce that there exist two real numbers $A$ and $B$ such that $$\forall x \in ] 0 , + \infty \left[ \quad z ( x ) = A \cos ( x ) + B \sin ( x ) + \lambda \int _ { x _ { 0 } } ^ { x } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } \right.$$
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ : $$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$ Let $x _ { 0 }$ in $] 0 , + \infty [$. We set for $x > 0$ $$h ( x ) = \int _ { x _ { 0 } } ^ { x } | z ( u ) | \frac { \mathrm { d } u } { u ^ { 2 } }$$ a) Show that there exist real constants $\mu$ and $M$ such that $h$ satisfies the differential inequality for $x \geqslant x _ { 0 }$ $$h ^ { \prime } ( x ) - \frac { \mu } { x ^ { 2 } } h ( x ) \leqslant \frac { M } { x ^ { 2 } }$$ Specify the constants $\mu$ and $M$ in terms of $A , B$ and $\lambda$. b) Deduce that $h$ is bounded on $\left[ x _ { 0 } , + \infty [ \right.$ and then that $z$ is bounded on the same interval. Multiply by $e ^ { \mu / x }$ and integrate the inequality from the previous question.
We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ : $$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$ Justify that $$\int _ { x } ^ { + \infty } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } = O \left( \frac { 1 } { x } \right)$$ near $+ \infty$. Deduce the existence of constants $\alpha$ and $\beta$ such that near $+ \infty$, $$z ( x ) = \alpha \cos ( x - \beta ) + O \left( \frac { 1 } { x } \right)$$
Let $n \in \mathbb { N }$. Show that there exists a pair of real numbers $( \alpha _ { n } , \beta _ { n } )$ such that for $x \rightarrow + \infty$, $$\varphi _ { n } ( x ) = \frac { \alpha _ { n } } { \sqrt { x } } \cos \left( x - \beta _ { n } \right) + O \left( \frac { 1 } { x \sqrt { x } } \right)$$
We introduce the differential equation $$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$ Using the bound from question II.E.2, show that $\varphi _ { 0 } ( 3 ) < 0$. Deduce that $\varphi _ { 0 }$ has a zero $\left. \alpha _ { 0 } \in \right] 0,3 [$. We admit that this is the first zero of $\varphi _ { 0 }$, that is, $\varphi _ { 0 }$ does not vanish on $] 0 , \alpha _ { 0 } [$.
We introduce the differential equation $$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$ Using question II.D, show by induction that for all integer $n \geqslant 1$ the function $\varphi _ { n }$ is strictly positive on $] 0 , \alpha _ { 0 } [$.
We introduce the differential equation $$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$ In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$. Let $a > A$. We set for $x > 0 , z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, solution of (IV.1). We define the function $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$. Verify that for $x > 0 , W ^ { \prime } ( x ) = \left( q ( x ) - c ^ { 2 } \right) z ( x ) z _ { 1 } ( x )$.
We introduce the differential equation $$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$ In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$. Let $a > A$, $z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, and $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$. We denote $\left. I _ { a } = \right] a , a + \pi / c \left[ \right.$ and assume that $\varphi _ { n }$ has no zeros on $I _ { a }$. Determine the signs of $W ( a ) , W ( a + \pi / c )$ and of $W ^ { \prime }$ on $I _ { a }$ and reach a contradiction. Deduce that $\varphi _ { n }$ has a zero in every interval $I _ { a }$ with $a > A$. One may distinguish cases according to the sign of $\varphi _ { n }$ on $I _ { a }$.