Algebraic functions

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Problem 5549 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.6 "Other Series Expansions"

Find the singular points of the equation $y^{3}-3y+2x = 0$ .

Remark.

Algebraic functions satisfy an equation $f(x,y) = 0$ , where $f(x,y)$ is a polynomial in the complex variables $x$ and $y$ . If both of these variables remain real, then the equation written represents an algebraic curve. If the equation has degree $n$ in $y$ , then it defines $n$ branches $y_{1}, y_{2}, \ldots , y_{n}$ of a multivalued function of $x$ . They represent regular functions of $x$ in a domain where their values are finite and unequal. If $x_{0}$ is a point lying in such a domain, then each of the functions $y_{\nu }$ expands in integer powers of $x-x_{0}$ . If, however, at the point $x_{0}$ several of the $y_{\nu }$ coincide, then in a neighborhood of this point the functions $y_{\nu }$ expand in fractional powers of $x-x_{0}$ by the formula

y = \sum _{\mu =\mu _{0}}^{\infty } a_{\nu }(x-x_{0})^{\frac{\mu }{m}},

where the number $\mu_{0}$ may be negative, and $m$ does not exceed $n$ . The values $a_{\nu }$ can be found by substituting these series into the equation. The singular points themselves can be found from the equations:

f(x,y) = 0, \quad \frac{\partial f}{\partial y} = 0.

Problem 5550 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.6 "Other Series Expansions"

Find the singular points of the equation $y^{3}-3y^{2}+x^{6} = 0$ .

Problem 5551 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.6 "Other Series Expansions"

Find the singular points of the equation $y^{3}-3y+2x^{2}(2-x^{2}) = 0$ .

Problem 5552 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 14.6 "Other Series Expansions"

Find the singular points of the equation $y^{3}-3y+2x^{3}(2-x^{3}) = 0$ .

Problem 4.166 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Various Propositions"

Let the function $w=\varphi (z)$ be regular (possibly many-valued) in the annular region $0 < \left|z\right| < \rho$ , $\rho > 0$ , and satisfy there identically an equation of the form

F_{0}(z) w^{l} + F_{1}(z) w^{l-1} + \cdots + F_{l-1}(z) w + F_{l}(z) = 0,

where $F_{0}(z), F_{1}(z), \ldots , F_{l-1}(z), F_{l}(z)$ are regular in a neighbourhood of the point $z=0$ . If there exists a power series $c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ such that the function $(\varphi (z)-c_{0}-c_{1}z-c_{2}z^{2}-\cdots -c_{n-1}z^{n-1})z^{-n}$ remains bounded in the neighbourhood of $z=0$ for infinitely many values of $n$ , then $\varphi (z)$ is regular in the neighbourhood of $z=0$ , and we have $\varphi (z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots +c_{n}z^{n}+\cdots$ .

Problem 4.169 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 1 "Various Propositions"

Let $f(z)$ be an analytic function, $z=x+iy$ , and let the square of the absolute value of $f(z)$

\left|f(z)\right|^{2} = \varphi (x,y)

be an algebraic function of the real variables $x$ and $y$ . Then the function $f(z)$ is itself an algebraic function of $z$ .

Remark.

H. A. Schwarz, oral communication.

Problem 2.16.2 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.16 "Analytic Continuation"

Suppose $(f, D)$ and $(g, D)$ are function elements, $P$ is a polynomial in two variables, and $P(f, g) = 0$ in $D$ . Suppose $f$ and $g$ can be analytically continued along a curve $\gamma$ , to $f_{1}$ and $g_{1}$ . Prove that $P\left(f_{1}, g_{1}\right) = 0$ . Extend this to more than two functions. Is there such a theorem for some class of functions $P$ which is larger than the polynomials?

References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.