Important Euler Number expressions

 I have presented a lot of expressions on Euler Number e. I have also given a model for it.

Today I am presenting three expressions. I might have written these expressions earlier. Or might not have. These expressions are of paramount importance to people in IT industry. I believe these are useful in developing logical ability. 

Have a very nice day, dear readers.

Three way Pi

Pi by 3 root 3 was the first Pi expression that I made. I have had reasons to make equations of Pi. Measuring behavior which was made very early in my life is the reason. Anyway, the first equation I got was when I followed Madhava of Sangamagrama.

Now I got three equations which give pi by 3 root 3.

Here it is

I hope you like them.

Expression of the day

I tried comb(2n,n) expressions. Many expressions on pi, square root of numbers and golden ratio were obtained. I used Wolfram Alpha to check(special thanks to Wolfram Alpha).

I tried Rational number series. Many expressions were obtained.

Today I decided to use both of them together. And what did I get ? Here we go . . . 

And here is the Wolfram Alpha screenshot.

I am deeply indebted to Wolfram Alpha. I mainly try mathematics only. Occasionally I also fire some Indian history queries too. It is amazing !!

Have a great day, dear blog readers !!

Back to back expressions

These are back to back expressions.

These are easy to verify. It is possibly 7th-8th class(India) material. Maybe some adults with not much of mathematics background may find them challenging. But I urge you to try them. You may have free time also due to COVID. Try these and verify.

But I feel the concept is BSc level. The concept of (mn+m-1)/(mn+m) - (mn-1)/(mn). Especially after so many (uncountable) results. Unfortunately they are not doctoral level. Otherwise I could have got a break. The gamma function result and the binomial results are advanced. But possibly only MSc level. But I am good in mathematics of electrical engineering. My basic qualification.

The expressions are 

Have a nice day, dear blog readers.

And now Gauss

Gauss made a brilliant equation. Integral of e to the power minus x squared for minus infinity to plus infinity equals square root of pi.

In place of x squared I put three consecutive numbers equation and four consecutive numbers equation. The result obtained is obedient. It is beautiful.

They are

I have used online Latex mathematical editor to type these.

Catalan's constant

I was working further on the complex function of mine.

I developed an expression. I tried it on Wolfram Alpha and the result show a term with Catalan's constant. It is indeed a privilege to get Catalan's constant in my expression.

Catalan's constant was first observed by Euler. However Charles Catalan did a lot of expressions and the constant is named after him. 

Below is the expression I made

Hooray !! A complex function !!

Made a complex function too. Here it is :

Have a nice day, dear blog readers !!

Hooray !! Integral and Sigma equivalence

 I have managed an integral and sigma equivalence. The equation is given below. The only other instance I know is from Parseval who did integral and sigma equivalence.

I am feeling ecstatic !!

Favorite four

 These are my favorite four expressions

You may try these for some random values and let me know if anything is wrong. I make errors inadvertently. So do let me know.

Have a nice day !!

Equations and COVID times

You take a look at the following.

If you have mathematical inclination then do the following.

1. Check for random value of n the validity of the four equations. These are approximately correct. You may put n as 17, 104 etc. 

2. Study the written text.

3. Ponder over the points written.

4. Kindly pass your comments.

5. Even if you check the four approximate expressions for values like 1, -1, 0, root 2 or i etc.; do write your comments.

My good wishes to you.

Two way reciprocals

 I am presenting below two expressions.

They are what I call "Two way reciprocals". The beauty of these two are unparalleled. They have a magnificent style and stands up with majesty like no other. 

 

Consider this equation

The other day I posted a blog post on Taylor series and Rational number series .

It is a great honor for me to have partly re-discovered this most amazing equation. As much as 70% of the world's system follow the Taylor series. It is that powerful.

Below I am pasting another expression. I feel it is damn important. And has unparalleled beauty as well. When I write damn important or unparalleled beauty I am not bragging. But I feel I am telling the truth.

Anyway forgive me if I hurt feelings due to the words in above paragraph.

The equation is

Sigma and Integral

The sigma operation came naturally to me. I made many expressions on sigma operations.

However, the integral operations did not happen. I worked on calculus yesterday night. I knew the basics. Then I managed to frame equations of similarity between sigma and integration. I took the help of Wolfram Alpha. Then I made expressions.

Here are the expressions.

Have a nice day, dear blog readers.

An expression on factorial

 Here I present an equation

I call it the walking man with arms and legs swaying in opposite ways. And its product gives factorial.

I hope you like it.

Have a nice day, dear blog readers.

Expressions on factorials

 Below I present expressions on factorials.

I hope you like them.

Have a nice day, dear blog readers.

Yet another finite expression

 Below I present three finite expressions on 1/(n+1). 

You may be aware of the following two infinite expressions on 1/(n+1)

But the one in top are finite expressions. Hope you like it.

Have a nice day, dear readers.

Equations at Ranchi

 

These equations were developed in Ranchi. They have been built from faith in God. I am just delivering his expressions. Just as no one realises that infinity is near to them; no one realises that God is still closer. 

I shall explain the significance:-

1. A finite sigma expression is damn tough. I do not know. Maybe infinite sigma expression is tougher. But I feel a finite sigma expression is tougher.

2. This combines the factorials with normal number in a mirror fashion.

3. This is an approximate expression that has been developed by sheer hard work. It took time. Understanding. But then it got developed.

4. I am taking the liberty to us comb(n,k). May I say it combines comb(n,k) with comb(k,1) in a beautiful way !!

This one is correct

In case there are errors in some of the equations, do let me know. Only this way can I correct myself.

Another expression

 I am presenting below another expression.

Approximate and Exact expressions

 

Expressions to stay cool during COVID-19

 COVID-19 is not going away. We must fight this pandemic by social distancing and maintaining and ensuring hygiene. It is better to be occupied with something.

Below are two expressions. You may verify them.

Have a nice day dear blog readers.