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2. Leonhard Euler
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3. Born 15 April 1707
Basel, Switzerland
Died 18 September 1783(aged 76)
[OS: 7 September 1783]
Saint Petersburg, Russian
Empire
Residence Kingdom of Prussia, Russian
Empire
Switzerland
Fields Mathematics and physics
Institutions Imperial Russian Academy of
Sciences
Berlin Academy
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4. • 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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6. Homothetic function
.allfor0iforallfor0is
thatmonotonic,strictlyisfunctiontheiffunction
homotheticaisthenfunction,shomogeneou
aisifThisfunctin.shomogeneoua
ofationtransformmontonicaisfunctionhomotheticA
21
yg'(y)yg'(y)
g(y)
g(y)z
),x,,xf(xy n


 
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7. 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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8. 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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9. 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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