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The complex plane and Euler's formula

z = x + iy as a point, modulus and argument, e^{iθ} = cos θ + i sin θ.

Complex numbers are points in the plane; multiplication rotates and scales. Euler's formula makes rotation an exponential, so e^{iπ} = −1. Picture it: polar form of 1 + i, the point at distance √2 and angle 45°. Think it: the exponential map wraps the imaginary axis around the unit circle; that periodicity is why the complex logarithm has branches.

ምሳሌ · e^(i*pi)

Arithmetic · Arithmetic

Evaluate e^(i*pi)

e^{i \pi}

እርምጃ በደረጃ

e^{i \pi} = -1
Power: E^(I·π) = -1.

መልሱን አሳይ

-1

አሁን ችግርን ይምረጡ፣ ወይም ጻፉ ወይም በራሳችሁ ቅርጽ አቅርቡ። እርምጃው ሁሉ፣ ምስል፣ መልሱ እስከምትጠይቅ ድረስ የተደበቀ ነው።

የራስዎን ስራዎች ይያዙ

ነጻ የሆኑት መተግበሪያዎች በየክፍል ውስጥ ማስታወሻዎችን ያካትታሉ፣ የተጠናቀቁትን ነገሮች መዝገብ፣ የተፈቱ ችግሮችን በአንድ ቦታ፣ ስለዚህ ገጽ መጠየቅ የሚችሉትን ረዳት ያካትታሉ፡፡ የቁጥር ትምህርት ለሁሉም ሰው ክፍት ነው፣ ተቀባይነት ያለው ወይም የለም፤

ምዝገባ መግባቱ

ፊደላት

ለሙሉ መግለጫ፣ ምስል፣ እና በውስጡ ያለውን ፊደል ትርጓሜ ለማየት ማንኛውንም ምልክት ይጫኑ።

እንዴት: The complex plane and Euler's formula

Power: E^(I·π) = -1.

ጥያቄዎች

Why is complex differentiability so much stronger than real?

The limit must be the same from every direction in the plane, not just two. That forces the Cauchy-Riemann equations, which in turn force infinitely many derivatives and a convergent Taylor series.

What is a residue?

The coefficient of 1/(z − a) in the Laurent series at a singularity a. The residue theorem says a contour integral equals 2πi times the sum of the residues inside, which evaluates many real integrals in one line.

በ Complex Analysis

Holomorphic functions and the Cauchy-Riemann equations Contour integrals and the residue theorem Power series and analytic continuation