A function f(x) is described as analytic around a point a, when f(x)  can be expressed as a power series ∑Cn (x-a)^n around a with a positive radius of convergence R.

This means that at any point c if |c-a|≤R, then the series  ∑Cn (c-a)^n converges.

Ex. f(x)=e^x  can be expressed as ∑((x-0)^n )/n! center of convergence 0.

By using ratio test on ∑(x^n)/n!, the radius of convergence is found to be R=∞

Then f(x)=e^x is an analytic function around 0.

What are the implications/uses of this function property?

If f(x) can be expressed as a power series around a point, then arithmetic operations can be made term by term in the series, as well as differentiation and integration. One use of  these operations  can be found when the power series of a given function cannot be described, but  the power series of the derivative or integral exists.

Ex. Evaluating ∫e^x² dx can be “easily” done by knowing

e^x=∑ (x^n) /n! which is analytic around 0 (from the previous example). Because it is analytical we can operate on it.

So: e^x²=∑ (x^2n) /n! =1+x²+(x^4)/2!+(x^6)/3!+…+x^(2n)/n!

And we can take the integral of every term in 1+x²+(x^4)/2!+(x^6)/3!+…+x^(2n)/n! ->  ∫e^x² dx

It is said that you learn more from your mistakes than from your successes. This thought can be applied in both personal and academic sides. As for today, I am going to focus on my past midterm.

Question 1 (f) was for me a challenging question. The reason for this is that this question requires not only knowing how to apply integration techniques, but also , as one of my instructor would say, it requires being a little bit smart. Overall, this exercice requires to join and wisely use almost all techniques studied in class.

A reasonable first aproach would be to use substitution method with what it’s inside the square root. However, that wouldn’t have been an effective approach. Looking carefully t²+1 could be substitute by using trigonometry equivalences. t= tan(θ). With this substituion we will be left with ∫cos(θ)/sin²(θ) dθ. Now, another integration technique is necessary: substitution. This method can be very useful when knowing how to make a good substitution.

Tips for the final:

• Practice is the key. Intuition when evaluating integrals comes from practice. Not all integrals with similar forms can be solved similarly but after practice there are patterns that you start to notice.
• Trigonometric substituions. Practice when and how to use trigonometric substitutions, how to change the limits of the integral, how far should I go with my substituions, and when to stop and use a different method.