Prove Holomorphic on C: Continuity and Differentiability

Thread starter fauboca
Start date Feb 21, 2012

In summary, the key idea is to find a holomorphic function g \colon \mathbb C \to \mathbb C that agrees with f everywhere except possibly on [2,5], and then use the Integral Transform Theorem to show that f and g agree on all points outside of C. However, we run into problems when trying to apply this to points inside C but not on [2,5], as the corollary to the theorem requires f to be holomorphic inside the circle. Therefore, alternative approaches such as Morera's theorem may need to be used to show that f is holomorphic at all points of C.

Feb 29, 2012

fauboca

Let ##\gamma## be a closed curve in ##\mathbb{C}##. If ##\gamma## doesn't contain any point from [2,5] in its interior, then ##\int_{\gamma}f=0## since f is holomorphic away from [2,5]. Suppose that ##\gamma## contains [2,5] in its interior. Let R be a rectangle oriented with the coordinate axes in ##\gamma## such that [2,5] is in R such that ##[2,5]\notin\partial R##. Divide the rectangle into two sub rectangles of length ##1-\epsilon## and ##\epsilon## such that [2,5] is contained in one the sub rectangles. WLOG suppose [2,5] is the rectangle of length ##\epsilon##. Then the over f of the rectangle of length ##1-\epsilon## is zero.

I am not sure how you mean to use the limit argument.

<h2>1. What does it mean for a function to be holomorphic on C?</h2><p>A function is considered holomorphic on C if it is complex differentiable at every point in the complex plane. This means that the function has a well-defined derivative at every point in the complex plane, which is a stronger condition than just being differentiable on the real line.</p><h2>2. How can I prove that a function is holomorphic on C?</h2><p>To prove that a function is holomorphic on C, you must show that it satisfies the Cauchy-Riemann equations. These equations relate the partial derivatives of the function's real and imaginary parts, and if they are satisfied, the function is complex differentiable and therefore holomorphic on C.</p><h2>3. Is a holomorphic function on C always continuous?</h2><p>Yes, a holomorphic function on C is always continuous. This is a consequence of the Cauchy-Riemann equations, which imply that the function's real and imaginary parts are continuously differentiable. Therefore, the function is continuous on the complex plane.</p><h2>4. Can a function be holomorphic on C but not differentiable at a specific point?</h2><p>No, a function cannot be holomorphic on C if it is not differentiable at a specific point. Holomorphicity requires complex differentiability at every point in the complex plane, so if a function is not differentiable at a specific point, it cannot be holomorphic on C.</p><h2>5. Are all analytic functions on C holomorphic?</h2><p>Yes, all analytic functions on C are holomorphic. This is because an analytic function is one that can be expressed as a convergent power series, and any function that can be represented as a power series is complex differentiable and therefore holomorphic on C.</p>

1. What does it mean for a function to be holomorphic on C?

A function is considered holomorphic on C if it is complex differentiable at every point in the complex plane. This means that the function has a well-defined derivative at every point in the complex plane, which is a stronger condition than just being differentiable on the real line.

2. How can I prove that a function is holomorphic on C?

To prove that a function is holomorphic on C, you must show that it satisfies the Cauchy-Riemann equations. These equations relate the partial derivatives of the function's real and imaginary parts, and if they are satisfied, the function is complex differentiable and therefore holomorphic on C.

3. Is a holomorphic function on C always continuous?

Yes, a holomorphic function on C is always continuous. This is a consequence of the Cauchy-Riemann equations, which imply that the function's real and imaginary parts are continuously differentiable. Therefore, the function is continuous on the complex plane.

4. Can a function be holomorphic on C but not differentiable at a specific point?

No, a function cannot be holomorphic on C if it is not differentiable at a specific point. Holomorphicity requires complex differentiability at every point in the complex plane, so if a function is not differentiable at a specific point, it cannot be holomorphic on C.

5. Are all analytic functions on C holomorphic?

Yes, all analytic functions on C are holomorphic. This is because an analytic function is one that can be expressed as a convergent power series, and any function that can be represented as a power series is complex differentiable and therefore holomorphic on C.