Explanation:

Euler's theorem is a fundamental theorem in mathematics that relates the exponential function with trigonometric functions. It is given by the equation e^(ix) = cos(x) + i*sin(x), where e is the base of the natural logarithm, i is the imaginary unit, x is a real number, and cos and sin are the cosine and sine functions, respectively.

Necessary condition of Euler's theorem:

The necessary condition for Euler's theorem is that z should be homogeneous and of order n. Homogeneous means that all the terms in the equation have the same degree, and order n means that the highest power of z in the equation is n. In other words, the equation should be of the form z^n = f(x,y), where f(x,y) is a function of x and y.

Explanation of necessary condition:

The reason for this necessary condition is that Euler's theorem is based on the Taylor series expansion of the exponential function. The Taylor series expansion of e^(ix) is given by the infinite sum of (ix)^n/n!, where n is a non-negative integer. By substituting z = cos(x) + i*sin(x) into this series, we obtain the equation z^n = e^(inx), which is Euler's theorem.

Therefore, in order for this substitution to be valid, z must be of the form cos(x) + i*sin(x), which is a homogeneous function of order 1. If z is not homogeneous or of a different order, then the substitution will not be valid, and Euler's theorem will not hold.

Conclusion:

In conclusion, the necessary condition for Euler's theorem is that z should be homogeneous and of order n. This condition ensures that the substitution of cos(x) + i*sin(x) into the Taylor series expansion of e^(ix) is valid, and Euler's theorem holds.