BeginnerTopic 24 of 104Engineering Mathematics
Euler's formula
e^{iθ}=cosθ+i sinθ links exponentials with trigonometry and underpins phasors and Fourier analysis.
In depth
Euler's formula e^{jθ}=cosθ+j sinθ links exponentials to trigonometry. Consequently cosθ=(e^{jθ}+e^{-jθ})/2. At θ=π it gives e^{jπ}+1=0. It underlies phasors, the Fourier transform and solving oscillator ODEs.
Key points
- Sinusoids are rotating complex exponentials
- Derivatives become multiplication by jω
- Simplifies trig identities
ExampleA voltage V cos(ωt+φ) is the real part of V e^{jφ}e^{jωt}.
What is Euler's formula?
e^{iθ}=cosθ+i sinθ links exponentials with trigonometry and underpins phasors and Fourier analysis.
Why is it important to understand euler's formula?
Euler's formula is the single identity that links exponentials with sines and cosines, and it is the reason phasor methods work.
Why is e to the power of i pi equal to minus one?
Setting theta equal to pi in Euler's formula gives cos pi plus i sin pi, which is minus one plus zero i, so e to the i pi equals minus one.
Do I need the exponential form in practice?
Usually yes. It turns differential equations for sinusoidal signals into algebra, which saves a great deal of work.
How is Euler's formula used in practice?
Typical applications include replacing a sinusoid by a complex exponential to simplify differentiation, describing a rotating phasor in a three-phase system and expressing the response of a damped system in closed form.
What should I learn before Euler's formula?
Study "Polar and exponential forms" first (topic 23 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Euler's formula?
Sinusoids are rotating complex exponentials. Derivatives become multiplication by jω. Simplifies trig identities.
Can you give an example of Euler's formula?
A voltage V cos(ωt+φ) is the real part of V e^{jφ}e^{jωt}.
What comes after Euler's formula?
The next topic is "Limits", which builds directly on this one.
How difficult is Euler's formula?
It is classed as Beginner in Engineering Mathematics (topic 24 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.