Question

Which one of the options given below is false?

• A. If \(f\) is analytic in a simply connected domain \(D\), then its line integral along any two different paths in \(D\) connecting the same endpoints \(P\) and \(Q\) are equal.
• B. There exists an entire function \(f\) such that \(f(1/n)=\dfrac{2n}{1+3n}\) for \(n\in\mathbb{N}\).
• C. If \(f\) is an entire function satisfying \(|f(z)|\le z^2\) for all \(|z|>1\), then \(f\) is a polynomial of degree at most \(2\).
• D. If \(T\) is a triangle with \(0\), \(1\) and \(i\) as its vertices, then \(\displaystyle\int_T z\,dz=0\).

Hint:

Test whether the values \(f(1/n)\) match a function with a pole, then apply the identity theorem.

Answer 1

The Correct Option is B

Step 1: Check option (A). This is the standard consequence of Cauchy's theorem: in a simply connected domain, an analytic function has a well-defined antiderivative, so its line integral is path-independent between fixed endpoints. This statement is true.

Step 2: Check option (B). Note that \(\dfrac{2n}{1+3n}=\dfrac{2}{3+1/n}\). Let \(g(z)=\dfrac{2}{3+z}\), analytic on \(\mathbb{C}\setminus\{-3\}\), with \(g(1/n)=\dfrac{2}{3+1/n}=\dfrac{2n}{1+3n}\), matching the required values. If an entire \(f\) satisfies \(f(1/n)=g(1/n)\) for all \(n\), then \(f\) and \(g\) agree on the sequence \(1/n\to0\), which accumulates inside the connected domain \(\mathbb{C}\setminus\{-3\}\) where both are analytic. By the identity theorem, \(f\equiv g\) on all of \(\mathbb{C}\setminus\{-3\}\). But as \(z\to-3\), \(g(z)\to\infty\) while \(f\), being entire, must stay finite at \(z=-3\) - a contradiction. So no such entire \(f\) exists, and statement (B) is false.

Step 3: Check option (C). This is a standard growth estimate result: an entire function bounded by a polynomial of degree \(2\) for large \(|z|\) must itself be a polynomial of degree at most \(2\) (via Cauchy's estimates on the Taylor coefficients). This statement is true.

Step 4: Check option (D). The function \(z\) is entire, hence analytic on and inside triangle \(T\). By the Cauchy-Goursat theorem, its integral over the closed triangular contour is \(0\). This statement is true.

Step 5: Only option (B) fails.

\[\boxed{\text{Option (B) is false}}\]