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Yashhttp://alltypeim.blogspot.in/

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Leonhard Euler

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Born 15 April 1707Basel, Switzerland

Died 18 September 1783(aged 76)[OS: 7 September 1783]Saint Petersburg, Russian Empire

Residence Kingdom of Prussia, Russian EmpireSwitzerland

Fields Mathematics and physicsInstitutions Imperial Russian Academy of S

ciences

Berlin Academy

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• Euler's formula, named after Leonhard Euler, is a mathematical formula incomplex analysis that establishes the fundamental relationship between thetrigonometric functions and the complex exponential function. Euler's formula states that, for any real number x:

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Homothetic function

. allfor 0 ifor allfor 0 is thatmonotonic,strictly is function theiffunction

homothetic a is then function, shomogeneoua is if This functin. shomogeneou aofation transformmontonic a isfunction homotheticA

21

yg'(y)yg'(y)g(y)

g(y)z),x,,xf(xy n

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Any function f(x) that possesses the characteristic mapping: is said to be homogeneous, with respect to x, to degree 1. By the same token, if f(x) obeys the mapping: then f(x) is homogeneous to degree “k”. In general, a multivariable function f(x1,x2,x3,…) is said to be homogeneous of degree “k” in variables xi(i=1,2,3,…) if for any value of λ,

For example, let us consider the function:

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How do we find out if this particular function is homogeneous, and if it is, to what degree? We evaluate this function at x=λx and y= λy to obtain:

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partial derivatives  , , and  . Hence, to complete the discussion on homogeneous functions, it is useful to study the mathematical theorem that establishes a relationship between a homogeneous function and its partial derivatives. This isEuler’s theorem.

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Thank you