Concerning Pi

Yesterday March 14 was dubbed Pi Day. It also happens that I started this website and blog one year ago ,and it’s Albert Einstein ‘s day of birth.

Here are some facts ,relations and formulas related to the number Pi.

Approximate value of Pi to 1500 digits (done with Mathematica);

pi-related-trig_3The Euler identity or formula ,which mathematicians like very much and find beautiful ,containing five math constants:

euler-pi-01Here is a quick derivation of a known formula involving Pi:

pi-related-trig_22where ln is the natural logarithm with base e = 2.7182818284590452354.

Two integrals involving Pi (done and verified with Mathematica):

pi-integAnd here is a brief study I made of a series of formulas involving Pi and the inverse trigonometric function arctangent (done with the Texas Instruments TI 92 Plus calculator and Mathematica)  :

pi001The general form for these relations  is:

pi000Or:

pi002A plot of the function funct-pi gives:

pi-related-trig01Using the Mathematica function FindRoot , we find two values for f(x) (with x >-1 and x< -1):

findroot-piThus we may deduce the following  values :

pi003

These pi formulas resemble or seem to be related to the Machin-Like Formulas or to the more general formulas :

pi-arctanwhich have been studied in the past.

I have also considered the study of other formulas involving Pi and other inverse trigonometric functions . Maybe I will publish or write more about these subjects in the future.

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