Let \(a_1,\ldots,a_n,b_1,\ldots,b_m\) be distinct complex numbers, let \(r_1,\ldots,r_n\) be nonnegative integers, and let \(c_1,\ldots,c_m\) be complex numbers. Prove that if \[ m\le r_1+\cdots+r_n+1, \] then there exists a rational function \(F(z)\in\mathbb C(z)\) satisfying all of the following:
\(F(z)\) is holomorphic at \(\infty\) and everywhere in \(\mathbb C\) except possibly at \(a_1,\ldots,a_n\).
\(\operatorname{ord}_{z=a_i}F(z)\ge -r_i\).
\(F(b_j)=c_j\) for \(j=1,\ldots,m\).
Solution
Set \[ R\coloneqq r_1+\cdots+r_n \] and \[ Q(z)\coloneqq\prod_{i=1}^n(z-a_i)^{r_i}. \]
1. There is a polynomial \(G\in\CC[z]\) with\[ \deg G\leq m-1\leq R \]such that\[ G(b_j)=c_jQ(b_j) \]for every \(j=1,\ldots,m\).
The points \(b_1,\ldots,b_m\) are pairwise distinct, so Lagrange interpolation gives a polynomial of degree at most \(m-1\) taking the prescribed values \[ c_jQ(b_j). \] The hypothesis \[ m\leq R+1 \] gives \(m-1\leq R\).
2. Define\[ F(z)\coloneqq\frac{G(z)}{Q(z)}. \]Then \(F\) is holomorphic on\[ \CC\setminus\{a_1,\ldots,a_n\} \]and at \(\infty\).
The only zeros of \(Q\) are among \(a_1,\ldots,a_n\), so the quotient is holomorphic away from those points.
Moreover, \[ \deg G\leq R=\deg Q. \] Hence the rational function \(G/Q\) has no pole at infinity. Equivalently, after writing \(w=1/z\), the function \[ F(1/w) \] extends holomorphically to \(w=0\) because multiplying numerator and denominator by \(w^R\) produces a quotient of polynomials in \(w\) whose denominator is nonzero at \(w=0\).
3. For every \(i=1,\ldots,n\),\[ \operatorname{ord}_{z=a_i}F(z)\geq-r_i. \]
Because the \(a_i\) are distinct, write \[ Q(z)=(z-a_i)^{r_i}Q_i(z), \] where \(Q_i(a_i)\neq0\). Then \[ (z-a_i)^{r_i}F(z) = \frac{G(z)}{Q_i(z)} \] is holomorphic at \(a_i\). This is exactly the assertion that \(\operatorname{ord}_{z=a_i}F\geq-r_i\).
4. For every \(j=1,\ldots,m\),\[ F(b_j)=c_j. \]
All of the \(a_i\) and \(b_j\) are distinct, so \(Q(b_j)\neq0\). By step 1, \[ F(b_j) = \frac{G(b_j)}{Q(b_j)} = \frac{c_jQ(b_j)}{Q(b_j)} = c_j. \]
5. Q.E.D.
Steps 2, 3, and 4 verify the three required properties of the rational function \(F\).